4.5 Logarithmic properties  (Page 6/10)

How does the power rule for logarithms help when solving logarithms with the form ${\log }_b\left(\sqrt[n]{x}\right)?$

What does the change-of-base formula do? Why is it useful when using a calculator?

${\log }_b\left(7x\cdot 2y\right)$

$\ln \left(3ab\cdot 5c\right)$

${\log }_b\left(\frac{13}{17}\right)$

${\log }_4\left(\frac{\frac{x}{z}}{w}\right)$

$\ln \left(\frac{1}{4^k}\right)$

${\log }_2\left(y^x\right)$

$\ln \left(7\right)+\ln \left(x\right)+\ln \left(y\right)$

${\log }_3(2)+{\log }_3(a)+{\log }_3(11)+{\log }_3(b)$

${\log }_b(28)-{\log }_b(7)$

$\ln \left(a\right)-\ln \left(d\right)-\ln \left(c\right)$

$-{\log }_b\left(\frac{1}{7}\right)$

$\frac{1}{3}\ln \left(8\right)$

$\log \left(\frac{x^{15}{y}^{13}}{z^{19}}\right)$

$\ln \left(\frac{a^{\mathrm{-2}}}{b^{\mathrm{-4}}{c}^5}\right)$

$\log \left(\sqrt{x^3{y}^{-4}}\right)$

$\ln \left(y\sqrt{\frac{y}{1-y}}\right)$

$\log \left(x^2{y}^3\sqrt[3]{x^2{y}^5}\right)$

$\log \left(2x^4\right)+\log \left(3x^5\right)$

$\ln (6x^9)-\ln (3x^2)$

$2\log (x)+3\log (x+1)$

$\log (x)-\frac{1}{2}\log (y)+3\log (z)$

$4{\log }_7\left(c\right)+\frac{{\log }_7\left(a\right)}{3}+\frac{{\log }_7\left(b\right)}{3}$

${\log }_7\left(15\right)$ to base $e$

${\log }_{14}\left(55.875\right)$ to base $10$

${\log }_{11}\left(5\right)$

${\log }_6\left(55\right)$

${\log }_{11}\left(\frac{6}{11}\right)$

${\log }_3\left(\frac{1}{9}\right)-3{\log }_3\left(3\right)$

$6{\log }_8\left(2\right)+\frac{{\log }_8\left(64\right)}{3{\log }_8\left(4\right)}$

$2{\log }_9\left(3\right)-4{\log }_9\left(3\right)+{\log }_9\left(\frac{1}{729}\right)$

${\log }_3\left(22\right)$

${\log }_8\left(65\right)$

${\log }_6\left(5.38\right)$

${\log }_4\left(\frac{15}{2}\right)$

${\log }_{\frac{1}{2}}\left(4.7\right)$

Use the product rule for logarithms to find all $x$ values such that ${\log }_{12}\left(2x+6\right)+{\log }_{12}\left(x+2\right)=2.$ Show the steps for solving.

Use the quotient rule for logarithms to find all $x$ values such that ${\log }_6\left(x+2\right)-{\log }_6\left(x-3\right)=1.$ Show the steps for solving.

Can the power property of logarithms be derived from the power property of exponents using the equation $b^x=m?$ If not, explain why. If so, show the derivation.

Prove that ${\log }_b\left(n\right)=\frac{1}{{\log }_n\left(b\right)}$ for any positive integers $b>1$ and $n>1.$

Does ${\log }_{81}\left(2401\right)={\log }_3\left(7\right)?$ Verify the claim algebraically.