Question Type 4: Using laws of logarithms Exercises

Solve for $x$: $\log_2(x) + \log_2(x - 3) = 3$.

Given $a=\log_2 3$ and $b=\log_2 5$, express $\log_2(45)$ in terms of $a$ and $b$.

Given $a=\log_2 3$ and $b=\log_2 5$, find $\log_2\sqrt{15}$ in terms of $a$ and $b$.

Simplify the expression $\log_{10}\!\Bigl(\dfrac{100x^2}{\sqrt{x}}\Bigr)$. [4 marks]

Solve for $x$: $\log_3(2x + 1) - \log_3(x - 1) = 2$.

Solve for $x$: $3\log_4(x) - \log_4(2) = 2$.

Expand the logarithm $\ln\!\Bigl(\dfrac{x^2\sqrt{y^3}}{z^4}\Bigr)$ into a sum and difference of simpler ln terms. [3 marks]

Given $\log_3(b) = 4$, find $\log_3\bigl(\tfrac{b}{9}\bigr)$.

Given $a=\log_5 2$ and $b=\log_5 3$, express $\log_5 72$ in terms of $a$ and $b$. [3]

Solve for $x$: $\ln(x) + \ln(x - 2) - \ln(3x) = 0$.

Given $\log_5 2=p$ and $\log_5 7=q$, express $\log_5\frac{14}{25}$ in terms of $p$ and $q$.

If $\log_3(b) = 4$, calculate $\log_3\bigl(1/b\bigr)$.

Given $a=\log_3 2$ and $b=\log_3 5$, express $\log_3 40$ in terms of $a$ and $b$.

If $\log_3(b) = 4$, determine $\log_3(27b)$.

Given $x, y, z > 0$, write the expression $\log\!\Bigl(\dfrac{\sqrt{x y^2}}{z^3}\Bigr)$ in terms of $\log x$, $\log y$ and $\log z$.

Simplify the expression $\log_3(81) + 2\log_3(3) - \log_3(9)$.

Given $a=\log_2 3$ and $b=\log_2 7$, express $\log_2\frac{21}{2}$ in terms of $a$ and $b$.

Solve for $x$: $\log_5(x^2 - 5x) = \log_5(6)$.

Given $\log_3(b) = 4$, find $\log_3\bigl(\sqrt{b}\bigr)$.

Given $a=\log_2 8$ and $b=\log_2 32$, express $\log_2 16$ in terms of $a$ and $b$.

Simplify the expression $\log_5(125) + 2\log_5\bigl(\tfrac{1}{5}\bigr)$.

Solve for $x$: $\log_3(x - 1) + \log_3(x + 2) = 2$.

Write the expression $2\ln(a) + 3\ln(b) - \ln(c)$ as a single logarithm.