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8.10: Solving Exponential Equations

Difficulty Level: At Grade Created by: CK-12
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Practice Solving Exponential Equations
 
 
 
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"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?

Guidance

Until now, we have only solved pretty basic exponential equations, like #1 in the Review Queue above. We know that \begin{align*}x=5\end{align*}, because \begin{align*}2^5=32\end{align*}. Ones like #4 are a little more challenging, but if we put everything 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. 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.

\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 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.

Intro Problem Revisit We can rewrite the equation \begin{align*}4^{x + 1} = 256\end{align*} as \begin{align*}2^{2(x+1)} = 2^8\end{align*} and solve for x.

\begin{align*}2^{2(x+1)} = 2^8\\ 2^{2x +2} = 2^8\\ 2x + 2 = 8\\ x = 3\end{align*}

Therefore, you're thinking of the number 3.

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 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*}

Practice

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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Vocabulary

TermDefinition
log "log" is the shorthand term for 'the logarithm of', as in "\log_b n" means "the logarithm, base b, of n."
Logarithm A logarithm is the inverse of an exponential function and is written \log_b a=x such that b^x=a.
take the log of both sides To take the log of both sides means to take the log of both the entire right hand side of the equation and the entire left hand side of the equation. As long as neither side is negative or equal to zero it maintains the equality of the two sides of the equation.

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