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Question

Welch Smith

07/04/2023 · Elementary School

Solve for x to the nearest 10th. 1350=660(1.075)^x+350

Answer
expertExpert-Verified Answer

Guerrero Garza
Competent Tutor
4.0 (40votes)

\(x\approx 5.7\)

Solution

  1. Start with the given equation:
    \[1350 = 660( 1.075) ^ x + 350\]
  2. Subtract 350 from both sides to isolate the term with the exponent:
    \[1350 - 350 = 660( 1.075) ^ x\]
    \[1000 = 660( 1.075) ^ x\]
  3. Divide both sides by 660 to further isolate the exponential term:
    \[\frac { 1000} { 660} = ( 1.075) ^ x\]
    \[\frac { 1000} { 660} \approx 1.5152\]
    \[1.5152 = ( 1.075) ^ x\]
  4. Take the natural logarithm (ln) of both sides to solve for \(x\):
    \[\ln ( 1.5152) = \ln ( ( 1.075) ^ x) \]
  5. Use the property of logarithms that \(\ln ( a^ b) = b \ln ( a) \):
    \[\ln ( 1.5152) = x \ln ( 1.075) \]
  6. Solve for \(x\):
    \[x = \frac { \ln ( 1.5152) } { \ln ( 1.075) } \]
  7. Calculate the values using a calculator:
    \[x \approx \frac { 0.4148} { 0.0728} \]
    \[x \approx 5.7\]
  8. Therefore, \(x \approx 5.7\) to the nearest tenth.

 

Supplemental Knowledge

Solving exponential equations often involves isolating the exponential term and then using logarithms to solve for the variable. This process can be broken down into several steps to simplify the equation.

 

Real-Life Connections

Exponential equations have numerous applications across numerous fields such as finance (compound interest), biology (population growth), and physics (radioactive decay). Learning to solve exponential equations allows one to make predictions and informed decisions using exponential models.

 

To master solving exponential equations and other algebraic concepts, explore UpStudy’s live tutor question bank or our AI-powered problem-solving services. Our resources provide detailed explanations and interactive learning experiences tailored to your needs.
For those looking to enhance their mathematical skills further, UpStudy offers comprehensive guides that make learning both engaging and effective!

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