# Solving Exponential Equations

## Equations with terms raised to exponents including x

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Solving Exponential Equations

Concept Extension: "I'm thinking of a number," you tell your best friend. "The number I'm thinking of satisfies the equation \begin{align*}4^{x + 1} = 256\end{align*}. What number are you thinking of?  Now can you verify that the number your best friend gave you is correct.  Show your algebraic steps!

Watch the following video on exponentials.

### Guidance

Until now, we have only solved pretty basic exponential equations. If we are solving the problem \begin{align*}{2}^{x} = 32\end{align*} then we know that \begin{align*}x=5\end{align*}, because \begin{align*}2^5=32\end{align*}. Others are a little more challenging. Consider the problem \begin{align*}{8}^{x}=128\end{align*}. If we put both values into a power of 2, we can set the exponents equal to each other and solve.

\begin{align*}8^x &= 128 \\ 2^{3x} &= 2^7 \\ 3x &= 7 \\ x &= \frac{7}{3}\end{align*}

So, \begin{align*}8^{\frac{7}{3}} = 128\end{align*}.

But, what happens when the power is not easily found? We must use logarithms, followed by the Power Property to solve for the exponent.

#### Example A

Solve \begin{align*}6^x=49\end{align*}. Round your answer to the nearest three decimal places.

Solution: To solve this exponential equation, let’s take the logarithm of both sides since the bases 6 and 49 can not be rewritten with the same base. The easiest logs to use are either \begin{align*}\ln\end{align*} (the natural log), or log (log, base 10). We will use the natural log.

\begin{align*}6^x &= 49 \\ \ln 6^x &= \ln 49 \\ x \ln 6 &= \ln 49 \\ x &= \frac{\ln 49}{\ln 6} \approx 2.172\end{align*}

#### Example B

Solve \begin{align*}10^{x-3}=100^{3x+11}\end{align*}.

Solution: Change 100 into a power of 10 in order to have the bases equal.

\begin{align*}10^{x-3} &= 10^{2(3x+11)} \\ x-3 &= 6x+22 \\ -25 &= 5x \\ -5 &= x\end{align*}

#### Example C

Solve \begin{align*}8^{2x-3}-4=5\end{align*}.

Solution: Add 4 to both sides since we most isloate the exponential and then take the log of both sides.

\begin{align*}8^{2x-3}-4 &= 5 \\ 8^{2x-3} &= 9 \\ \log 8^{2x-3} &= \log 9 \\ (2x-3)\log 8 &= \log 9 \\ 2x-3 &= \frac{\log 9}{\log 8} \\ 2x &= 3 + \frac{\log 9}{\log 8} \\ x &= \frac{3}{2}+\frac{\log 9}{2 \log 8} \approx 2.56\end{align*}

Notice that we did not find the numeric value of \begin{align*}\log9\end{align*} or \begin{align*}\log8\end{align*} until the very end. This will ensure that we have the most accurate answer.

### Guided Practice

Solve the following exponential equations.

1. \begin{align*}4^{x-8}=16\end{align*}

2. \begin{align*}2(7)^{3x+1} =48\end{align*}

3. \begin{align*}\frac{2}{3} \cdot 5^{x+2}+9=21\end{align*}

1. Change 16 to \begin{align*}4^2\end{align*} and set the exponents equal to each other.

\begin{align*}4^{x-8} &= 16 \\ 4^{x-8} &= 4^2 \\ x-8 &= 2 \\ x &=10\end{align*}

2. Divide both sides by 2 to isolate the exponential and then take the log of both sides.

\begin{align*}2(7)^{3x+1} &= 48 \\ 7^{3x+1} &= 24 \\ \ln 7^{3x+1} &= \ln 24 \\ (3x+1)\ln 7 &= \ln 24 \\ 3x+1 &= \frac{\ln 24}{\ln 7} \\ 3x &= -1 + \frac{\ln 24}{\ln 7} \\ x &= -\frac{1}{3} + \frac{\ln 24}{3 \ln 7} \approx 0.211\end{align*}

3. Subtract 9 from both sides and multiply both sides by \begin{align*}\frac{3}{2}\end{align*}. Then, take the log of both sides.

\begin{align*}\frac{2}{3} \cdot 5^{x+2}+9 &= 21 \\ \frac{2}{3} \cdot 5^{x+2} &= 12 \\ 5^{x+2} &= 18 \\ (x+2)\log 5 &= \log 18 \\ x &= \frac{\log 18}{\log 5}-2 \approx -0.204\end{align*}

### Homework

Use logarithms and a calculator to solve the following equations for \begin{align*}x\end{align*}. Round answers to three decimal places.

1. \begin{align*}5^x = 65\end{align*}
2. \begin{align*}7^x = 75\end{align*}
3. \begin{align*}2^x = 90\end{align*}
4. \begin{align*}3^{x-2} = 43\end{align*}
5. \begin{align*}6^{x+1}+3=13\end{align*}
6. \begin{align*}6(11^{3x-2})=216\end{align*}
7. \begin{align*}8+13^{2x-5}=35\end{align*}
8. \begin{align*}\frac{1}{2} \cdot 7^{x-3}-5=14\end{align*}

Solve the following exponential equations without a calculator.

1. \begin{align*}4^x=8\end{align*}
2. \begin{align*}9^{x-2} = 27\end{align*}
3. \begin{align*}5^{2x+1}=125\end{align*}
4. \begin{align*}9^3=3^{4x-6}\end{align*}
5. \begin{align*}7(2^{x-3})=56\end{align*}
6. \begin{align*}16^x \cdot 4^{x+1}=32^{x+1}\end{align*}
7. \begin{align*}3^{3x+5}=3 \cdot 9^{x+3}\end{align*}

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