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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*}

Answers

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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