Question on logarithm with variable base

Question

The question is $4.7=x^.25$ At first I tried to write it in log form: $\log_x4.7=.25$ Then I realized I can just raise the original equation to the fourth power. But, going with my initial approach, how would I finish the problem? Is there a calculator that can solve a log with a variable in the base? Thanks, Aaron

Answer 1

Realize that $$\log_{x}\frac{47}{10}=\frac{1}{4}$$ This implies that $$\log_{\frac{47}{10}}x=4$$ Because $$\log_{a}b \times \log_{b}a=1$$ So we have that $$x=\left(\frac{47}{10}\right)^4$$ So you can see using logarithims provides the same result, albeit in a more roundabout way. Also, to answer your second question, most CASIO calculators and Wolframalpha can solve a log with a variable in the base.

Answer 2

Logarithms are not required to solve this problem. Look at the position of the $x$ in your problem. It is raised to a power hence this should be solved similar to other power problems you have encountered such as: $x^3=8$. You would not solve $x^3=8$ using logarithms so you shouldn't for your problem either.

$$x^{0.25}=4.7$$

$$x=4.7^\frac{1}{0.25}$$

$$x=4.7^4$$

Contrast this to a problem where you have have to use logarithms such as $2^x=32$. In this problem the $x$ is in the exponent position. The difference here in the placement of the $x$ in the question is what tells you what technique to use in solving the problem.

Answer 3

You want to isolate the $x$. Putting $x$ in $\log_x$ does nothing to isolate $x$ and gets you no closer. (It's stepping sideways really... and making an average problem hard.)

To isolate the $x$ is $x^{.25} = 4.7$ you must eliminate the power. And to do that is to take it to the inverse power.

$x^{.25} = 4.7$

$(x^{.25})^{1/.25} = 4.7^{1/.25}$

$x = 4.7^{\frac 1{\frac{1}{4}}} = 4.7^4$.