Logarithms with variables

Question

What is a,

$$\log_{\sqrt 3} \sqrt[6]a,$$

If $\log_a 27=b$

Can anyone help me with this?? A tip or a hint will be super helpful too!! The answer should include b ..

Answer 1

First, we need to take $a$ out of the base, and make all the bases the same, which is usually the first step in these types of problems. Using the change of base property, we turn $\log_{a}27$ into $\frac{\log_{\sqrt{3}}27}{\log_{\sqrt{3}}a}$. All we need to do now, is manipulate this into the desired form. Starting with $\frac{\log_{\sqrt{3}}27}{\log_{\sqrt{3}}a}=b$, we simplify the top logarithm to get $\frac{6}{\log_{\sqrt{3}}a}=b$. Dividing by b and multiplying by $\log_{\sqrt{3}}a$, we get $\frac{6}{b}=\log_\sqrt{3}a$. Dividing by 6, we get $\frac{1}{b}={\log_\sqrt{3}a}*\frac{1}{6}$. Using the power property on the left side, we now have $\frac{1}{b}=\log_\sqrt{3}\sqrt[6]{a}$.

Answer 2

On one hand we have:

$\log_{\sqrt{3}}(\sqrt[6]{a})=\dfrac{\ln(\sqrt[6]{a})}{\ln(\sqrt{3})}=\dfrac{\ln(a^{1/6})}{\ln(3^{1/2})}=\dfrac{\frac 16\ln(a)}{\frac 12\ln(3)}=\dfrac{\ln(a)}{3\ln(3)}=\dfrac{\ln(a)}{\ln(3^3)}=\dfrac{\ln(a)}{\ln(27)}$

On the other hand:

$b=\log_a(27)=\dfrac{\ln(27)}{\ln(a)}$

Answer 3

An alternative expression of the standard rules for logarithms$^*$:

$$\log_y(x)=\frac{1}{log_x(y)}\\\log_y(x)=\log_z(x)\log_y(z),$$ from which is straightforward to see that $$\log_{y^m}(x^n)=\frac{n}{m}\log_y(x),$$

allows one to argue here as follows: $$\log_{\sqrt3}\sqrt[6]{a}\,=\,\frac{1/6}{1/2}\,\log_3a\,=\,\frac{1}{3}\frac{1}{\log_a3}\,=\,\frac{1}{3}\frac{3}{b}\,=\,\frac{1}{b}$$

Answer 4

Here's a hint:

$\log _{a} b = c$

is equivalent to

$a^c = b$

which is also equivalent to

$\sqrt[1/c]{a} = b$

Try rearranging the formulas in the question, possibly changing all of the equations to the form $a^b = c$.