Do you know logarithmic rules?

Algebra Level 3

Let $a$ and $b$ be two positive numbers different from 1, such that $a^{3}\not =b^{2}$ , and $x=\log_{\frac{a^3}{b^2}}{\frac{b}{a}}$ . Then, which of the following expressions is $a^{3x+1}$ equal to?

b^{2x+1}

b^{x}

a*b

1

e^{x*y}

b^{2x-1}

1 solution

Arturo Presa
Aug 13, 2015

Using the change-of-base formula we get $x=\frac{\ln \frac{b}{a}}{\ln \frac{a^{3}}{b^{2}}}=\frac{\ln b - \ln a}{3\ln a - 2\ln b}$ Multiplying both sides by $3\ln a - 2\ln b$ $x(3\ln a - 2\ln b)=\ln b - \ln a$ Transposing terms, we obtain the equivalent equation $3x \ln a +\ln a=2x \ln b+\ln b$ or $(3x+1) \ln a =(2x+1) \ln b$ Thus, $\ln a^{3x+1} = \ln b^{2x+1}$ Therefore the answer must be $b^{2x+1}.$

Even simpler:

$x=\log_{\frac{a^3}{b^2}}{\frac{b}{a}}$ $\Rightarrow$ $\left(\frac{a^3}{b^2}\right)^x=\frac{b}{a}$ by the properties of logarithm which further simplifies to $\boxed{a^{3x+1}=b^{2x+1}}$ by grouping like terms.

Scott Ripperda - 5 years, 10 months ago

Yes, this is simpler! Thank you for sharing it!

Arturo Presa - 5 years, 10 months ago

Do you know logarithmic rules?

1 solution

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