Exponential Equations and Inequalities - Pre-Calculus

Card 1 of 40

0

Didn't Know

0

Didn't Know

Knew It

0

1 of 2019 left

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

Simplify the log expression:

Tap to reveal answer

Answer

The logarithmic expression is as simplified as can be.

← Didn't Know|Knew It →

Question

Solving an exponential equation.

Solve for ,

$5^{x}=250$ .

Tap to reveal answer

Answer

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

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

← Didn't Know|Knew It →

Question

Solving an exponential equation.

Solve

$10^{3x}=100$

Tap to reveal answer

Answer

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

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

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

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

← Didn't Know|Knew It →

Question

Solve

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

Tap to reveal answer

Answer

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 .

← Didn't Know|Knew It →

Question

Solve the equation for using the rules of logarithms.

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

Tap to reveal answer

Answer

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.

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

← Didn't Know|Knew It →

Question

Solve the following equation:

Tap to reveal answer

Answer

To solve this equation, recall the following property:

Can be rewritten as

$log_{10}175=x$

Evaluate with your calculator to get

$x\approx2.24$

← Didn't Know|Knew It →

Question

Solve for x:

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

Simplify the log expression:

Tap to reveal answer

Answer

The logarithmic expression is as simplified as can be.

← Didn't Know|Knew It →

Question

Solving an exponential equation.

Solve for ,

$5^{x}=250$ .

Tap to reveal answer

Answer

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

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

← Didn't Know|Knew It →

Question

Solving an exponential equation.

Solve

$10^{3x}=100$

Tap to reveal answer

Answer

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

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

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

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

← Didn't Know|Knew It →

Question

Solve

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

Tap to reveal answer

Answer

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 .

← Didn't Know|Knew It →

Question

Solve the equation for using the rules of logarithms.

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

Tap to reveal answer

Answer

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.

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

← Didn't Know|Knew It →

Question

Solve the following equation:

Tap to reveal answer

Answer

To solve this equation, recall the following property:

Can be rewritten as

$log_{10}175=x$

Evaluate with your calculator to get

$x\approx2.24$

← Didn't Know|Knew It →

Question

Solve for x:

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

0

Knew It