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Exponential Functions: (lesson 2 of 3)

## Exponential Equations

### Very easy equation:

Example 1: Solve for $x$ in the equation

$3^x=9$

Solution:

\begin{aligned} 3^x &= 9 \\ 3^x &= 3^2 \\ x &= 2 \end{aligned}

### Not so easy equations:

It is often necessary to use a logarithm when solving an exponential equation.

Example 2:

Solve for $x$ in the equation $3^x = 10$

Solution:

Step 1: Take the natural logarithm of both sides:

$\ln {3^x} = \ln 10$

Step 2: Use the rule $\ln {a^x} = x \ln a$.

$x \ln 3 = \ln 10$

Step 3: Find $x$:

$x = \frac{{\ln 10}}{{\ln 3}} \approx \frac{{2.302585}}{{1.098612}} \approx 2.0959$

Example 3:

Solve for $x$ in the equation $e^x = 10$

Solution:

Step 1: Take the natural logarithm of both sides:

$\ln{e^x} = \ln10$

Step 2: Use the rule $\ln{a^x} = x \ln{a}$.

$x \ln{e} = \ln10$

Step 3: Use the rule $\ln{e} = 1$

$x = \ln10$

Step 4: Find $x$

$x = \ln10 \approx 2.302585$

Example 4:

Solve for $x$ in the equation $10^{x+5} - 3 = 2$

Solution:

Step 1:Isolate the exponential term

$10^{x+5}-3=2$

$10^{x+5}=5$

Step 2: Take the natural logarithm of both sides:

$\ln{10^{x+5}} = \ln5$

Step 3: Use the rule $ln{a^x} = x \ln{a}$.

$(x+5) \ln10 = \ln5$

Step 4: Find $x$:

$x = \frac{\ln5}{\ln10}-5$

Example 5:

Solve for $x$ in the equation $3^{2x} - 5 \cdot 3^x + 6 = 0$

Solution:

Step 1: Rewrite the equation in quadratic form:

${(3^x)}^2 - 5 \cdot 3^x + 6 = 0$

Step 2: Factor the left side of the equation ${(3^x)}^2 - 5 \cdot 3^x + 6 = 0$.

${(3^x)}^2 - 5 \cdot 3^x + 6 = (3^x - 3) (3^x - 2)$

Step 3: Solve $3^x - 3 = 0$ and $3^x - 2 = 0$