Use Logarithms To Solve Exponential Equations and Inequalities

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Pre-Calculus › Use Logarithms To Solve Exponential Equations and Inequalities

Questions 1 - 10

1

Simplify the log expression:

Cannot be simplified any further

Explanation

The logarithmic expression is as simplified as can be.

2

Solving an exponential equation.

Solve for ,

$5^{x}=250$ .

$x=3+log_{5}2$

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

Explanation

We recall the property:

$log_b(x)=y \leftrightarrow b^y=x$

$x=log_{5}(125\cdot 2)=log_{5}125+log_{5}2$

Now, $log_{5}125=log_{5}(5^{3})=3$ .

Thus

$x=3+log_{5}2$ .

3

Solve the equation for using the rules of logarithms.

$\ln(17x)+\ln\left(\frac{x}{3}\right)+\log_3(x^2)=4$

$x\approx 1.04$

$x\approx 75.15$

$x\approx1.86$

Explanation

Expanding the logarithms into sums of logarithms will cancel out the first two x terms, resulting in the equation:

$\ln(17)+ln(3)+2\log_3(x)=4$

Combining the first and second terms, then subtracting the new term over will allow you to isolate the variable term.

Divide both sides of the equation by 2, then exponentiate with 3.

Evaluating this term numerically will give the correct answer.

$2\log_3(x)=4-\ln(51)\implies \log_3(x)=2-\frac{\ln(51)}{2}\ \ \implies x=3^{2-\ln(51)/2}\approx1.04$

4

Solve

$\frac{e^{x^2-2}}{e^{3x}}=e^{2x-6}$ .

Explanation

After using the division rule to simplify the left hand side you can take the natural log of both sides.

If you then combine like terms you get a quadratic equation which factors to,

.

Setting each binomial equal to zero and solving for we get the solution to be .

5

Solve for x in the following equation: $3^{5x-3}=9^{2x}$

Explanation

$3^{5x-3}=3^{(2)(2x)}\rightarrow 5x-3=4x\rightarrow x=3$

6

Solve the following equation:

$x\approx2.24$

$x\approx4.48$

$x\approx22.4$

$x\approx0.224$

Explanation

To solve this equation, recall the following property:

Can be rewritten as

$log_{10}175=x$

Evaluate with your calculator to get

$x\approx2.24$

7

Solve for x: $\log_{4}(3x-2)=2$

Explanation

$log_4 (3x-2)=2\rightarrow 3x-2=4^2\rightarrow 3x=18\rightarrow x=6$

8

Solve for x using the rules of logarithms: $\log_{3}2x-\log_{3}(x-3)=1$

Explanation

$log_3 \frac{2x}{x-3}=1\rightarrow 3^{log_3 \frac{2x}{x-3}}=3^1\rightarrow \frac{2x}{x-3}=3\rightarrow x=9$

9

Solve for x:

Explanation

$3^x=3^4\rightarrow x=4$

10

Solving an exponential equation.

Solve

$10^{3x}=100$

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

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

Explanation

Use (which is just $log_{10}$ , by convention) to solve.

$log(10^{3x})=log(100)$

$log(10^{3x})=log(10^2)$

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

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