## Using the Definition of a Logarithm to Solve Logarithmic Equations

In this unit, you will explore the techniques for solving logarithmic equations. We will begin by using the definition of a logarithm to "undo" it. Then, we will work up to more complex techniques.

### Using the Definition of a Logarithm to Solve Logarithmic Equations

We have already seen that every logarithmic equation $log_b(x)=y$ is equivalent to the exponential equation $b^y=x$. We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.

For example, consider the equation $log_2(2)+log_2(3x−5)=3$.  To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for 5$x$:

$log_2(2)+log_2(3x−5)=3$
$log_2(2(3x−5))=3$
Apply the product rule of logarithms.
$log_2(6x−10)=3$
Distribute.
$2^3=6x−10$
Apply the definition of a logarithm.
$8=6x−10$
Calculate $2^3$.
$18=6x$
$x=3$
Divide by 6.

#### Using the Definition of a Logarithm to Solve Logarithmic Equations

For any algebraic expression $S$ and real numbers $b$ and $c$, where $b > 0$ , $b≠1$,

$log_b(S)=c$ if and only if $b^c=S$

#### Example 9

Using Algebra to Solve a Logarithmic Equation

Solve $2lnx+3=7$.

Solution

$2lnx+3=7$
$2lnx=4$
Subtract 3.
$lnx=2$
Divide by 2.
$x=e^2$
Rewrite in exponential form.

##### Try It #9

Solve $6+lnx=10$.

#### Example 10

Using Algebra Before and After Using the Definition of the Natural Logarithm

Solve $2ln(6x)=7$.

Solution

$2ln(6x)=7$
$In (6x) = \dfrac{7}{2}$
Divide by 2.
$6x = e^{(\dfrac{7}{2})}$
Use the definition of $ln$.
$x = \dfrac{1}{6} e^{(\dfrac{7}{2})}$
Divide by 6.

##### Try It #10

Solve $2ln(x+1)=10$.

#### Example 11

Using a Graph to Understand the Solution to a Logarithmic Equation

Solve $ln x=3$.

Solution

$ln x=3$

$x=e^3$  Use the definition of the natural logarithm.

Figure 3 represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20. In other words $e^3≈20$. A calculator gives a better approximation: $e^3≈20.0855$.

Figure 3 The graphs of $y=ln x$ and $y=3$ cross at the point $(e^3,3)$, which is approximately (20.0855, 3).

##### Try It #11

Use a graphing calculator to estimate the approximate solution to the logarithmic equation $2^x=1000$ to 2 decimal places.

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-6-exponential-and-logarithmic-equations