Solving Logarithmic Equations

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Math › Solving Logarithmic Equations

Questions 1 - 10

Solve the equation.

$10^{2x-10}=(\frac{1}{100})^{8x-1}$

$x=\frac{2}{3}$

$x=\frac{1}{3}$

$x=-\frac{2}{3}$

Explanation

Change the right side to $10^{-2}$ so that both sides have the same bsae of 10. Apply log and then set the exponential expressions equal to each other

Solve the equation.

$5^{3x-9}=25^{^{x+5}$

Explanation

First, change 25 to so that both sides have the same base. Once they have the same base, you can apply log to both sides so that you can set their exponents equal to each other, which yields .

Solve the equation.

$7^{3x+4}=49^{x+1}$

Explanation

Change 49 to so that both sides have the same base so that you can apply log. Then, you can set the exponential expressions equal to each other .

Thus,

Solve the equation.

$3^{3x-7}=81^{12-3x}$

$x=\frac{11}{3}$

$x=\frac{22}{3}$

Explanation

Change 81 to so that both sides have the same base. Once you have the same base, apply log to both sides so that you can set the exponential expressions equal to each other (). Thus, $x=\frac{11}{3}$ .

Solve the equation.

$2^{9x-3}=\left(\frac{1}{32}\right)^{x-11}$

$x=\frac{29}{7}$

$x=\frac{22}{7}$

$x=\frac{22}{14}$

$x=\frac{1}{9}$

$x=-\frac{26}{7}$

Explanation

Change the right side to $2^{-5}$ so that both sides are the same. Apply log to both sides so that you can set the exponential expressions equal to each other ().

$x=\frac{29}{7}$

Solve the equation.

$4^{2x-7}=64^{3x}$

Explanation

Change 64 to so that both sides have the same base. Apply log to both sides so that you can set the exponential expressions equal to each other

Thus, .

Solve the equation.

$8^{x-1}=(\frac{1}{16})^{4x-3}$

$x=\frac{15}{19}$

$x=\frac{11}{19}$

Explanation

Change the left side to and the right side to $2^{-4}$ so that both sides have the same base. Apply log and then set the exponential expressions equal to each other (). Thus, $x=\frac{15}{19}$ .

Solve the equation.

$5^{11-3x}=125^{5-x}$

No solution

Explanation

Change 125 to so that both sides have the same base. Apply log and then set the exponential expressions equal to each other so that . Upon trying to isolate , it becomes clear that there is no solution.

Solve the equation.

$36^{10x+2}=(\frac{1}{6})^{13-x}$

$x=-\frac{17}{19}$

$x=\frac{17}{19}$

$x=\frac{11}{19}$

Explanation

Change the left side to and the right side to $6^{-1}$ so that both sides have the same base. Apply log to both sides and then set the exponential expressions equal to each other (). $x=-\frac{17}{19}$ .

Solve the equation.

$25^{20x+4}=(\frac{1}{125})^{4-3x}$

$x=-\frac{20}{31}$

$x=\frac{20}{31}$

$x=\frac{9}{40}$

$x=-\frac{11}{31}$

Explanation

Change the left side to and the right side to $5^{-3}$ so that both sides have the same base. Apply log and then set the exponential expressions equal to each other (). Thus, $x=-\frac{20}{31}$

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