# Test: Logarithms II - Challenging

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Question 1:   Find $x$: $2={\mathrm{log}}_{8}x$
$64$
$49$
$16$
$256$
Question 2:   Find $x$: $x={\mathrm{log}}_{3}243$
$5$
$81$
$27$
$9$
Question 3:   Solve ${\mathrm{log}}_{2}8+{\mathrm{log}}_{8}2$
$16$
$3\frac{1}{3}$
$4$
$1$
Question 4:   Express ${\mathrm{log}}_{7}x$ as multiple of ${\mathrm{log}}_{10}x$
${\mathrm{log}}_{10}x-{\mathrm{log}}_{3}x$
${\mathrm{log}}_{10}x{\mathrm{log}}_{3}x$
$\frac{{\mathrm{log}}_{10}x}{{\mathrm{log}}_{3}x}$
${\mathrm{log}}_{10}x{\mathrm{log}}_{7}10$
Question 5:   Express ${\mathrm{log}}_{4}17$ in terms of the logarithms base $10$
$17{\mathrm{log}}_{10}4$
${\mathrm{log}}_{10}17/{\mathrm{log}}_{10}4$
$4{\mathrm{log}}_{10}17$
${\mathrm{log}}_{10}4/{\mathrm{log}}_{10}17$
Question 6:   Simplify $3{\mathrm{log}}_{a}2x+2{\mathrm{log}}_{a}{x}^{2}-6{\mathrm{log}}_{a}x$
$-{\mathrm{log}}_{a}\left(x-{x}^{2}\right)$
$2{\mathrm{log}}_{a}x$
${\mathrm{log}}_{a}6{x}^{2}$
${\mathrm{log}}_{a}8x$
Question 7:   Simplify $2{\mathrm{log}}_{a}\frac{1}{2}x-{\mathrm{log}}_{a}3x+2{\mathrm{log}}_{a}2x$
${\mathrm{log}}_{a}\frac{3}{4}{x}^{3}$
${\mathrm{log}}_{a}\frac{4}{3}{x}^{2}$
${\mathrm{log}}_{a}\left(\frac{{x}^{3}}{3}\right)$
$3{\mathrm{log}}_{a}x$
Question 8:   Find $x$: $6{\mathrm{log}}_{a}x\frac{1}{3}-{\mathrm{log}}_{a}2x=2{\mathrm{log}}_{a}x$
$x=1$
$x=2$
$x=\frac{1}{2}$
$x=\sqrt{2}$
Question 9:   Find $x$: ${12}^{{\mathrm{log}}_{12}5}=x$
$x=4$
$x=12$
$x=5$
$x=\frac{1}{5}$
Question 10:   Find $y$ in terms of $x$: $2{\mathrm{log}}_{a}x+3{\mathrm{log}}_{a}y=3{\mathrm{log}}_{a}2x+3{\mathrm{log}}_{a}\sqrt{y}$
$y=4\sqrt[3]{{x}^{2}}$
$y=\frac{1}{2}{x}^{4}$
$y=3\sqrt{{x}^{5}}$
$y=\frac{1}{4}{x}^{3}$