The Exponential and Logarithmic Forms of an Equation

Logarithmic equations can be written as exponential equations and vice versa. The logarithmic equation $\log_b(x)=c$ corresponds to the exponential equation $b^{c}=x$.

As an example, the logarithmic equation $\log{_2}16=4$ can be converted to the exponential equation $2^4=16$.

The logarithmic equation $\log_4(64)=3$ can be converted into the exponential equation $4^3=64$.

Solving Logarithmic Equations

Conversion from logarithmic to exponential form can help one solve otherwise difficult equations.

Example 1

Solve for $x$ in the equation $\log{_3}243=x$ 

Here we are looking for the exponent to which $3$ is raised to yield $243$. It might be more familiar if we convert the equation to exponential form giving us: 

$3^x=243 \\3^5=243$

The exponent we seek is $5$. Thus, $\log{_3}243=5$.

The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation. Starting with $243$, if we take its logarithm with base $3$, then raise $3$ to the logarithm, we will once again arrive at $243$.

Example 2

Solve for $x$ in the equation $\log_6(x-2)=3$

If we write the logarithmic equation as an exponential equation we obtain:

$6^{\log_6(x-2)}=6^3$ 

As the exponent and log on the left side of the equation undo each other we are left with: 

$\begin{aligned} x-2&=6^3 \\x-2&=216 \\x&=218 \end{aligned}$

Solving Exponential Equations

An exponential equation is an equation where the variable we are solving for appears in the exponent. 

If the equation consists of two terms set equal to each other and these terms have the same base, then the exponents are equal. We can use this fact to solve such exponential equations as follows:

Example 3 

Solve for $x$ in the equation $5^{5x+8}=5^{x^{2}+3x}$

Here since the bases are both $5$, the exponents are equal. We use this fact to solve the equation as follows:

 $\begin{aligned} 5x+8&=x^2+3x \\ 0&=x^2+3x-5x-8 \\ 0&=x^2-2x-8 \\ 0&=(x-4)(x+2) \end{aligned}$

$x=4 \text{ or } x=-2$

Example 4 

Solve for x in the equation $3^{x+1}=81^x$

Here the bases are not equal, but it is possible to write 81 using a base of 3 as follows:

 $\begin{aligned} 3^{x+1}&=(3^4)^x \\ 3^{x+1}&=3^{4x} \end{aligned}$

At this point, the left and right sides of the equation have the same base so we can solve for $x$ by setting the two exponents equal to each other:

$\begin{aligned} x+1&=4x \\ 1&=4x-x \\ 1&=3x \\ x&=\frac{1}{3} \end{aligned} $

Solving Exponential Equations Using Logarithms

In many cases, an exponential equation cannot be solved by using the methods of example $3$ and $4$ above because the bases cannot easily be made equal. In these cases taking the logarithm of both sides of the equation allows us to solve the equation. While you can take the log to any base, it is common to use the common log with a base of $10$ or the natural log with the base of $e$. This is because scientific and graphing calculators are equipped with a button for the common log that reads $log$, and a button for the natural log that reads $ln$ which allows us to obtain a good approximation for the common or natural log of a number.

Example 5

Solve for $x$ in the equation $2^x=17$

Here we cannot easily write $17$ with a base of $2$ so instead we take the log of both sides as follows.

$\log{2^x}=\log17$

Next we use the properties of logarithms to move the variable out of the exponent.

$x\log2=\log17$

Lastly we divide by $\log2$ to solve for $x$.

$x=\frac{\log17}{\log2} $

It is important to note that this is an exact answer. We can arrive at an approximation by using the $\log$ button on your calculator.

$x=\frac{\log17}{\log2}\approx4.0877 $

Example 6

Solve for $x$ in the equation $2^x=17$ using the natural log

Here we will use the natural logarithm instead to illustrate the fact that any base will do.

$\begin{aligned} \ln{2^x}&=\ln17 \\ x\ln2&=\ln17 \\x&=\frac{\ln17}{\ln2}\approx4.0877 \end{aligned} $

Example 7 

Solve for $x$ in the equation $3=4^{5x+18} $

Again, we use logarithms to move the variable out of the exponent allowing us to solve for x as follows:

$\begin{aligned} \log3&=\log4^{5x+18} \\ \log3&=(5x+8)\log4 \\ \frac{\log3}{\log4}&=5x+8 \\ \frac{\log3}{\log4}-8&=5x \\ \frac{\log3}{5\log4}-\frac{8}{5}&=x \\ \frac{\log3-8\log4}{5\log4}&=x \end{aligned}$

Now we can use the properties of logarithms to re-write the left hand side and solve for $x$:

$\dfrac{\log\frac{3}{4^8}}{\log4^5}=\dfrac{\log\frac{3}{65536}}{\log1024}\approx-1.4415$