IB Mathematics Analysis and Approaches Question Type 7: Using Change of Base Formula Exercises

Simplify the expression $\frac{3}{{\log }_2(x)}+{\log }_x(y)$ 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 $x>0$: $${\log }_2(x)+{\log }_4(x)+{\log }_8(x)=6.$$

If ${\log }_3(b)=4$, evaluate ${\log }_{27}(b)$.

Express ${\log }_5(7)$ in terms of natural logarithms.

Simplify ${\log }_3(16)+\frac{2}{{\log }_4(3)}$ into a single quotient of natural logarithms.

Solve for $x>0$, $x\ne 1$: $${\log }_x(81)+\frac{4}{{\log }_3(x)}=6$$

Solve for $x>0$, $x\ne 1$: $$\frac{3}{{\log }_2(x)}+{\log }_x(16)=5.$$

Show that for $a>0$, $b>0$, $a\ne 1$, $b\ne 1$: $${\log }_a(b){\log }_b(a)=1.$$

Express ${\log }_4(27)$ in terms of ${\log }_2(3)$.

Evaluate the expression $$\frac{3}{{\log }_2(8)}+\frac{2}{{\log }_4(16)}$$

Express ${\log }_8(18)$ in terms of ${\log }_2(3)$.

Simplify ${\log }_2(9)-\frac{2}{{\log }_3(2)}$ into an expression involving $\ln 2$ and $\ln 3$.