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Take the logarithm of both sides of the equation. We'll use the common logarithm (10).

Mathematics

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Answer

n=2.97n = 2.97

Step-by-step solution

8 steps

Here's how to solve the equation:

  1. Solve the equation, (0.3)n+1=(0.2)n(0.3)^{n+1} = (0.2)^n to two decimal places.

  1. Take the logarithm of both sides of the equation.

    We'll use the common logarithm (log⁡10\log_{10}).

    log⁡10((0.3)n+1)=log⁡10((0.2)n)\log_{10}((0.3)^{n+1}) = \log_{10}((0.2)^n)

  2. Apply the logarithm power rule, log⁡b(xp)=plog⁡bx\log_b(x^p) = p \log_b x, to both sides.

    (n+1)log⁡10(0.3)=nlog⁡10(0.2)(n+1)\log_{10}(0.3) = n\log_{10}(0.2)

  3. Distribute log⁡10(0.3)\log_{10}(0.3) on the left side.

    nlog⁡10(0.3)+log⁡10(0.3)=nlog⁡10(0.2)n\log_{10}(0.3) + \log_{10}(0.3) = n\log_{10}(0.2)

  4. Gather all terms containing nn on one side and constant terms on the other side.

    nlog⁡10(0.3)−nlog⁡10(0.2)=−log⁡10(0.3)n\log_{10}(0.3) - n\log_{10}(0.2) = -\log_{10}(0.3)

  5. Factor out nn from the terms on the left side.

    n(log⁡10(0.3)−log⁡10(0.2))=−log⁡10(0.3)n(\log_{10}(0.3) - \log_{10}(0.2)) = -\log_{10}(0.3)

  6. Solve for nn by dividing both sides by (log⁡10(0.3)−log⁡10(0.2))(\log_{10}(0.3) - \log_{10}(0.2)).

    n=−log⁡10(0.3)log⁡10(0.3)−log⁡10(0.2)n = \frac{-\log_{10}(0.3)}{\log_{10}(0.3) - \log_{10}(0.2)}

  7. Calculate the numerical values of the logarithms and then nn.

    Using a calculator:
    log⁡10(0.3)≈−0.5228787\log_{10}(0.3) \approx -0.5228787
    log⁡10(0.2)≈−0.6989700\log_{10}(0.2) \approx -0.6989700

    Substitute these values into the equation for nn:

    n=−(−0.5228787)(−0.5228787)−(−0.6989700)n = \frac{-(-0.5228787)}{(-0.5228787) - (-0.6989700)} n=0.52287870.6989700−0.5228787n = \frac{0.5228787}{0.6989700 - 0.5228787} n=0.52287870.1760913n = \frac{0.5228787}{0.1760913} n≈2.96939n \approx 2.96939

  8. Round the value of nn to two decimal places.

    n≈2.97n \approx 2.97

    The solution is n=2.97n = 2.97.

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