Omni Calculator logo
Board

How to Solve Log Equations Like a Pro

If you just got your homework and are faced with several log equations to solve, do not get scared. We are here to help you master the process of finding solutions for them. There are some math equations that involve logarithms, which are popularly known as log equations. These equations can be easily solved by applying the log rules. You can explore each one of the rules by accessing our article titled "Log Rules Made Simple: Understanding the Laws of Logarithms".

The three main approaches for solving logarithmic equations are:

  • Set the arguments equal if the logs have the same base;
  • Convert log to exponential form; and
  • Use the change-of-base formula to convert logs to the same base.

We will cover all these different methods in detail and with examples.

Moreover, in this article, you will see the following subjects:

  • Solving logarithmic equations: setting the arguments equal;
  • How to solve log equations by converting log to exponential;
  • How to solve logarithmic equations using the change-of-base formula;
  • Solving logs using a calculator;
  • How to solve log3 243;
  • How to solve log equations without a calculator;
  • And more.

So, get ready to apply the log rules and solve your log equations.

Would you like to know how to solve log equations without a calculator? Well, this is not the purpose of Omni Calculator, but you are in the right article. As we pointed out, there are three types of log equations. Let us look at the first of them in detail. If you have an equation where all the logs have the same base, you can use the setting the arguments equal method. This is considered the simplest type of log equation, since an algebraic equation replaces the initial expression. Let us see it clearly with the following example:

log⁡a(M)=log⁡a(N)M=N\begin{split} & \log_a(M) = \log_a(N)\\[1em] & M = N \end{split}

You can add some complexities to this method by applying extra log properties, such as the product rule:

log⁡a(M)=log⁡a(N)+log⁡a(O)log⁡a(M)=log⁡a(N⋅O)M=N⋅O\begin{split} & \log_a(M) = \log_a(N)+\log_a(O) \\[1em] & \log_a(M) = \log_a(N \cdot O) \\[1em] & M = N \cdot O \end{split}

Here we used the product rule to write log⁡a(N)+log⁡a(O)=log⁡a(N⋅O)\log_a(N)+\log_a(O) = \log_a(N \cdot O) , enabling us to find the final result.

Now, let's see how to apply this method to determine the solution of a linear equation. Suppose that log⁡3(2x−2)+log⁡3(1/2)=log⁡3(3−x)\log_3(2x-2)+\log_3(1/2) = \log_3(3-x), then, what is the value of xx? This exercise can be solved by taking the following steps:

log⁡3(2x−2)+log⁡3(1/2)=log⁡3(3−x)log⁡3((2x−2)⋅1/2)=log⁡3(3−x)log⁡3(x−1)=log⁡3(3−x)2 x=3+1x=42x=2\small \begin{split} & \log_3(2x-2)+\log_3(1/2) = \log_3(3-x) \\[1em] & \log_3((2x-2)\cdot 1/2) = \log_3(3-x) \\[1em] & \log_3(x-1) = \log_3(3-x) \\[1em] & 2\,x = 3+1 \\[1em] & x = \frac{4}{2} \\[1em] & x = 2 \end{split}

Suppose that you have an equation where your log is equal to a constant. Then, you can convert this log to an exponential to determine the final answer. This procedure is mathematically described by:

log⁡a(M)=CM=aC\begin{split} & \log_a(M) = C \\[1em] & M = a^C \end{split}

And how can I solve log equations with natural logarithms? In this case, we just need to change the base aa for ee, or in other words:

ln⁡(M)=BM=eB\begin{split} & \ln(M) = B \\[1em] & M = e^B \end{split}

What happens if your logs have different bases? This type of equation is a little bit more complex than our previous examples. In order to solve them, we need to use the change-of-basis formula, which is given by:

log⁡a(M)=log⁡b(M)log⁡b(a)\log_{a}(M) = \frac{\log_b(M)}{\log_b(a)}

We can see this formula in practice in the following example:

log⁡2(x)=log⁡5(9)ln⁡(x)ln⁡(2)=ln⁡(9)ln⁡(5)ln⁡(x)=ln⁡(9)ln⁡(5)⋅ln⁡(2)x=eln⁡(9)ln⁡(5)⋅ln⁡(2)x=2.57\begin{split} & \log_2(x) = \log_5(9) \\[1em] & \frac{\ln(x)}{\ln(2)} = \frac{\ln(9)}{\ln(5)} \\[1em] & \ln(x) = \frac{\ln(9)}{\ln(5)} \cdot \ln(2) \\[1em] & x = e^{\frac{\small\ln(9)}{\small\ln(5)}\small \cdot \ln(2)} \\[1em] & x = 2.57 \end{split}

We know you are probably tired after solving all these equations by hand. Well, we can help you with some interesting tools to simplify the computation of your logarithms, enabling you to solve the log equations more easily and quickly. Take a look at our dedicated tools:

Feel free to check them and learn how to become a log master.

You can find the correct answer for this log equation by following the steps below:

  1. Factorize 243 in powers of 3;

    243 = 35

  2. Apply the following rules for exponents and logs:

    M = ay and loga M = y

  3. Compute the final answer:

    log3 243 = 5

For solving logs with different bases, you need to use the change-of-bases formula, which is written as:

loga(M) = logb(M)/logb(a)

The base b can have any value, but we generally choose to work with b=10 or b=e, for common and natural logarithms, respectively.

If you do not have a calculator, you can follow the steps below to solve log equations properly:

  1. Identify the base and the argument of each log.
  2. Use the log rules to simplify your equation.
  3. Verify if you need to set the arguments equal, convert log to exponential form, or use change-of-base formula.
  4. Simplify the resulting mathematical operations.
  5. If you have a variable, isolate it in order to solve your problem.

This article was written by João Rafael Lucio dos Santos and reviewed by Steven Wooding.