Question Type 7: Using change of base formula Exercises

Simplify the expression $\displaystyle \frac{3}{\log_2(x)} + \log_x(y)$ into a form involving $\ln x$ and $\ln y$.

Simplify the expression $\frac{2}{\log_5(7)}-\log_7(25)$ by converting to a form involving natural logarithms.

Solve for $x>0$:

If $\log_3(b)=4$, evaluate $\log_{27}(b)$.

Express $\log_5(7)$ in terms of natural logarithms.

Simplify $\displaystyle \log_3(16) + \frac{2}{\log_4(3)}$ into a single quotient of natural logarithms.

Solve for $x>0$, $x \neq 1$:

Solve for $x>0$, $x\neq1$: $\frac{3}{\log_2(x)}+\log_x(16)=5.$

Show that for $a > 0$, $b > 0$, $a \neq 1$, $b \neq 1$: $\log_a(b) \, \log_b(a) = 1.$

Express $\log_4(27)$ in terms of $\log_2(3)$.

Evaluate the expression

Express $\log_8(18)$ in terms of $\log_2(3)$.

Simplify $\displaystyle \log_2(9) - \frac{2}{\log_3(2)}$ into an expression involving $\ln 2$ and $\ln 3$.