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Chapter 5: Problem 59

Solve. $$9^{x-1}=100\left(3^{x}\right)$$

Short Answer

Expert verified
x = 2 + \frac{2}{\log_{10}(3)}

Step by step solution

01

Rewrite the Equation

Rewrite the given equation: \[9^{x-1} = 100(3^x)\]. Note that 9 can be written as \(3^2\), so the equation becomes: \[(3^2)^{x-1} = 100(3^x)\].
02

Simplify the Exponential Expression

Use the power rule of exponents to simplify \((3^2)^{x-1}\) to \(3^{2(x-1)}\). The equation now becomes: \[3^{2(x-1)} = 100(3^x)\].
03

Equate the Exponents

Since both sides of the equation have a base of 3, set the exponents equal to each other:\[2(x-1) = x + \log_3(100)\].
04

Solve for x

Distribute the 2 on the left side:\[2x - 2 = x + \log_3(100)\]. Move all x terms to one side and constants to the other side:\[2x - x = \log_3(100) + 2\]. Simplify:\[x = 2 + \log_3(100)\].
05

Evaluate Solution

Convert \(\log_3(100)\) to a common logarithm using the change of base formula: \[x = 2 + \frac{\log_{10}(100)}{\log_{10}(3)}\]. Since \log_{10}(100) = 2, the equation becomes: \[x = 2 + \frac{2}{\log_{10}(3)}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithms
Logarithms are a way of expressing exponential equations in a different form. They help us solve equations where the unknown is an exponent. The logarithm \(\text{log}_b(a)\) answers the question: 'To what exponent must the base \(b\) be raised to produce \(a\)?' For example, \( \text{log}_3(100) \) asks '3 raised to what power equals 100?' Understanding this concept is critical to solving complex exponential equations.
Exponents
Exponents are used to represent repeated multiplication of a number by itself. For instance, \(3^2 = 3 * 3 = 9\). Exponents follow specific rules:
  • Power Rule: \( (a^m)^n = a^{mn} \)
  • Product Rule: \( a^m * a^n = a^{m+n} \)
  • Quotient Rule: \( a^m / a^n = a^{m-n} \)
These rules help simplify complex exponential expressions, as seen in the exercise.
Change of Base Formula
The change of base formula is a useful tool for evaluating logarithms with bases other than 10 or e. It states: \[ \text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)} \] When solving the equation in our exercise, we used the change of base formula to convert \( \text{log}_3(100) \) to common logarithms: \[ \text{log}_3(100) = \frac{\text{log}_{10}(100)}{\text{log}_{10}(3)} \] This allowed us to evaluate the logarithm using a calculator.
Solving Equations
To solve equations, follow a logical process to isolate the variable. In the given exercise, we:
  • Rewrote the equation using properties of exponents.
  • Simplified the expressions.
  • Equated the exponents with a common base.
  • Used the change of base formula to solve for \( x \).
This systematic approach helps break down complex problems into manageable steps.

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Most popular questions from this chapter

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