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## How to Solve Logarithmic Equations

In this video there are two problems modeled. The first example works with common logs and the second example models an equation with a natural log. The keys to solving any type of logarithmic equation: being able to write the given equation in exponential form or being able to take the log of each side.

To be able to write any logarithmic function in exponential form knowing the definition of logarithm and natural logarithm is key. So here they are…

Definition: Logarithm – The log to the base b of a positive number y i s defined:

If $y = b^x$, then $log_b \hspace{0.1 cm}y = x$.

Definition: Natural Logarithm – If $y = e^x$ , then $log_e \hspace{0.1 cm}y = x$ which is also written as $ln \hspace{0.1 cm}y = x$. The natural logarithmic function is the inverse, written as $y = ln \hspace{0.1 cm}x$.

In other words, if $y = e^x$, then $y = ln \hspace{0.1 cm}x$.

Now that you have reviewed the definition of log and natural log, read through these two examples, then watch the embedded algebra 2 video.

### Example 1 Solving a Logarithmic Equation

This a problem that I assigned to my Algebra 2 class for a review before a quiz on solving logarithmic equations:

$7^{x-3} = 25$ Given

$log 7^{x-3} = log 25$ Take the log of each side

$(x-3)log 7 = log 25$ Product Property

$x-3 = \frac{log25}{log7}$ Division Property

$x = \frac{log25}{log7}+3$ Addition Property

$x \approx 4.6542$ Use a calculator

As you can see in this example, you can easily solve an equation with multiple log in it. All you need to do is use the properties of logarithms and that taking the log of each side is a legal mathematical move. The properties for common logs are used with natural logs.

### Example 2 Solving a Natural Logarithmic Equation

Solve the following equation.

$2 \cdot ln \hspace{0.1 cm}x + 3 \cdot ln \hspace{0.1 cm} 2 = 5$ Given

$ln \hspace{0.1 cm}x^2 + ln \hspace{0.1 cm}2^3 = 5$ Power Property

$ln \hspace{0.1 cm}x^2 + ln \hspace{0.1 cm}8 = 5$ Simplify

$ln \hspace{0.1 cm}8x^2 = 5$ Product Property

$8x^2 = e^5$ Write in Exponential Form

$x^2 = \frac{e^5}{8}$ Division Property

$x = \sqrt{ \frac{e^5}{8}}$ Square Root Property

$x \approx 4.3072$ Use a calculator

Solving logarithmic equations with natural log is easy if you can use the properties of natural logs and you know how to write a natural logarithmic equation in exponential form.