Ever wondered how to reverse an exponent? That’s where logarithms step in! In this article, we’ll dive into the key logarithm formulas and why they’re super important for solving tricky math problems—especially on exams like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP tests, PERT, Accuplacer, and even the MCAT.

So, what’s a logarithm? Think of it as the math tool that helps you figure out exactly what exponent you need to get a certain number. Basically, logarithms are the inverse of exponential functions. If exponents are all about multiplying numbers repeatedly, logarithms undo that process—they tell you how many times you need to multiply a base number to reach a target value.

And here’s something cool: logarithms aren’t just abstract math concepts. They pop up everywhere—from analyzing data trends to solving real-life growth and decay problems. Plus, when you understand how logarithms connect inputs to outputs (just like other functions), you'll feel much more confident tackling questions in the math section of any standardized test.

Ready to crack the code of logs? Let’s break it down!

Logarithmic Functions

The logarithmic function equation is as shown, \(c=\log_{b}a\) for a>0 such that b>0 and \(b\ne1\). This is read as “log a to the base b is equal to c” or “c is equal to the log a to the base b”. Also, if \(b^c = a\) then only we can define \(log_{b}a = c\). Mathematically it means, to what power b must be raised, to yield a. For example for the equation \(2^x= 5\), the value of x can be obtained by converting this exponent form to logarithmic form i.e. \(x = log_25\).

For the equation \(c=\log_ba\)

• The first condition for c to exist is that the base must be positive. i.e. b > 0. This is due to the reason that negative numbers cannot be raised to all exponents.
• The second condition for c to exist is base can’t be 1. i.e. b ≠ 1 because 1 raised to any power will give 1. So, \(log_{1} 5\) is not defined.
• Thirdly for x to exist a should be positive i.e a > 0.

Some of the common bases of the log function are base 2, base 10 and base e.

Logarithmic functions are a crucial tool for arithmetic computations and find many uses in large-number astronomical and scientific computations. Logarithmic functions are thought of as the inverse of exponential functions and have a tight relationship with them. The logarithmic function logaN = x is obtained by transforming the exponential function ax = N.

Domain and Range of Log Functions

Let's examine the fundamental logarithmic function f(x) = log x (or y = log x). Log x is valid only for x > 0. Attempting to find log 0, log (-1), log (-2), etc., on your calculator results in an error, highlighting the domain as the set of all positive real numbers. Now, let's explore various x-values (inputs) to observe the corresponding y-values (outputs) of the function.
When x = 1, y = log 1 = 0
When x = 2, y = log 2 = 0.3010
When x = 0.2, y = -0.6990
When x = 0.01, y = -2, etc
As we can see, y can be either a positive or negative real number (or zero). Y can, therefore, have any real number value. The group of all real numbers is, hence, the range of a logarithmic function. 
Thus:
The domain of log function y = log x is x > 0 (or) (0, ∞).
The range of any log function is the set of all real numbers (R)

Types of Logarithmic Functions

There are two types of functions which are logarithmic functions : common logarithms or the standard logarithms(also known as base 10 logarithms). The second one is the natural logarithm which holds a base as e.

Standard Logarithm

When the base of a log operator is 10 it is called a standard logarithm denoted as: \(f(x)=\log_{10}x\)

This is also read as a common logarithm. Sometimes a log equation is written without any base, as shown below:

\(x=\log_{ }a,\ where\ a>0\), in such an equation the base by default is equal to 10.

Natural Logarithm

Similar to the common log function, when the base of a log operator is e it is called a natural logarithm denoted as: \(x=\log_ea,\ or\ \ln\ a,\ where\ a\ >0\), wherein e=2.718281828……

Therefore you can confirm that whenever you see a log function as; ln(x), remember that it is the log function with base e i.e. \(log_{e} x\).

Properties of Logarithmic Functions

Well acknowledged with the definition and types, let us step towards the different related properties.

How to Solve Logarithmic Functions

The logarithmic function can be solved using the various logarithmic formulas such as; the power rule, product rule, quotient rule, equality rule and change of base rule. All these rules are discussed with the related formula previously. Similarly, the logarithm functions can also be solved by switching them to exponential form. The natural log or ln is the inverse function of e. That implies one can undo the other one i.e.

\(\begin{array}{l}\ln(e^x)=x\\
e^{\ln x}=x\end{array}\)

Logarithmic Functions Graph

The mathematical formula for the logarithmic functions is; \(f\left(y\right)=\log_b\left(y\right)\). Let us now understand how to graph logarithmic functions. Depending on the base the function can be decreasing; when the value of the base lies between 0 to 1 or the function can be increasing; when the value of the base is greater than 1. function. The graph of the different log functions is displayed below.

Derivatives of Logarithmic Functions

Familiar with the different properties and how to solve the log function, learn the formula for the derivatives of logarithmic functions in the below table.

Difference between Logarithmic and Exponential Functions

The logarithmic functions are said to be inverse of exponential functions. Consider \(y=\log_bx\) represents a logarithmic function then the exponential function is given by \(x=b^y\).

Solved Examples for Logarithmic Functions

Functions like logarithmic, exponential, and trigonometric functions are best understood with the help of solved examples. Below are some examples with solutions based on the definition, properties and different rules. Try them on your own and verify your answer with the solutions.

Solved Example 1: If \(\log_{7}343=z\), then find z.

Solution: Given,\(\log_{7}343=z\), this can be written as \(7^z=343\)

The reason is that log functions are the opposite of exponential functions.

Hence, if \(7^z=343\), then z=3. As the cube of 7 is equal to 343.

Solved Example 2: If \(\log_3\left(\frac{1}{27}\right)=z\), then find the value of z.

Solution: Given; \(\log_3\left(\frac{1}{27}\right)=z\)

The equation becomes \(3^z=\left(\frac{1}{27}\right)\)

Also, \(\frac{1}{27}=3^{-3}\Rightarrow3^z=3^{-3}\)

Therefore z=-3.

Solved Example 3: If \(\log_2\left(-8\right)=z\), then determine the value of z?

Solution: Given that; \(\log_2\left(-8\right)=z\)

The equation when transformed to exponential function becomes \(2^z=-8\)

This is not possible, as logarithms have only a positive domain. Hence, the result cannot be specified.

Solved Example 4: If \(\ln\left(e^2\right)=z\), obtain the value of z?

Solution: If the base is not specified then consider as base e

The equation becomes \(e^z=e^2\), after taking antilog on both sides.

i.e \(z=2\)

\(\ln\left(e^2\right)=2\)

Solved Example 5: Obtain the value of log 20 + log 3 using the logarithmic function rules?

Solution: Using the product rule, \(\log_b\left(m\times n\right)=\log_bm+\log_bn\), we can reframe the equation as:

log 20 + log 3=\(\log(20\times3)\)

log 20 + log 3=log 60.

Solved Example 6: Determine the value of \(\log_6\left(36\right)-\log_6\left(6\right)\)?

Solution: Given, \(\log_6\left(36\right)-\log_6\left(6\right)\)

Using the division property; \(\log_b\left(\frac{m}{n}\right)=\log_bm-\log_bn\)

\(\log_6\left(36\right)-\log_6\left(6\right)=\log_6\left(\frac{36}{6}\right)\)

=\(\log_6\left(\frac{36}{6}\right)=\log_6\left(6\right)=1\)

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Questions to Ace Your Exams

Q1. Which is the graph of a logarithmic function? How to graph logarithmic functions?  
The graph of a logarithmic function has a distinct shape, and it is characterized by a curve that approaches but never touches the x-axis (the line y=0). For the purpose of graphing a logarithmic function, a set of general steps can be adhered to:
Begin by identifying the base and key points, encompassing intercepts, asymptotes, and any other points of significance.
Generate a table of values by selecting different x values and computing their corresponding y values using the logarithmic function.
Proceed to plot the acquired points.
Sketch the asymptotes. In cases where the base surpasses 1, the asymptote manifests as a vertical line at x=0. Alternatively, if the base resides between 0 and 1, the asymptote is vertical and occurs at x=∞.
Finally, delineate the curve, bearing in mind the characteristic behavior where it approaches but refrains from intersecting the x-axis.

Q2. What are logarithmic functions? How to do logarithmic functions? How to find the derivative of logarithmic functions?    
A logarithmic function is the inverse of an exponential function. Follow these things for logarithmic functions: 
If the unknown exists outside of the log, assess the log. Most logs require the use of a calculator to assess.
If the unknown is included inside the log, transform it to exponential form.
If an equation has many logs, attempt to consolidate as many as feasible.
To find the derivative of a log, first write the log in its power form: a^y = x. Then, use implicit differentiation and take the derivative of each side with respect to x. Lastly, solve for dy/dx. The result is the derivative of the original log.

Q3. How to find the domain of a logarithmic function? How to find the inverse of logarithmic functions?    
A logarithmic function's domain may be found by looking at the values that the logarithmic expression, usually denoted by the variable x, is valid for. Logarithmic functions can be calculated in their inverse by converting the function to base e on both sides. Calculate the inverse of logarithmic functions by transforming the function into base e on both sides. Convert it into an exponential equation with base e, isolate the term, and examine the function's domain. Confirm your result by assessing symmetry with the line y=x.

Q4. How to find asymptotes of logarithmic functions? How to find vertical asymptotes of logarithmic functions?
You can find any logarithmic function's vertical asymptote by setting its input to zero. A logarithmic function's vertical asymptotes can be found by taking into account the values of x at which the logarithmic expression turns undefined. Vertical asymptotes happen when the logarithm's argument gets close to 0 or negative. The vertical asymptote of a logarithmic function of the form y=log b (x), where b is the logarithm's base, may be obtained by solving for x using log b (x) and putting the argument within the logarithm equal to zero

Conclusion

Mastering logarithms can seriously boost your confidence when tackling math problems, especially on standardized tests like the SAT, ACT, or GRE. With a solid grasp of log rules and properties, solving exponential equations becomes much easier. Keep in mind, consistent practice is key to getting comfortable with these concepts. So take your time, work through examples, and soon enough, logarithmic functions won’t feel intimidating at all!