Solving Exponential Equations - Math

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Question

What are the x-intercepts of this equation?

Answer

To find the x-intercepts, set the numerator equal to zero.

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Question

Solve the equation for .

$\small $9^x$$=3^6$$

Answer

Begin by recognizing that both sides of the equation have a root term of .

$\small $9^x$$=3^6$$

Using the power rule, we can set the exponents equal to each other.

$\small 2x=6$

$\small x=3$

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Question

Solve the equation for .

$\small $3^{2x}$=81$

Answer

Begin by recognizing that both sides of the equation have the same root term, .

$\small $3^{2x}$=81$

$\small $3^{2x}$$=9^2$$

We can use the power rule to combine exponents.

Set the exponents equal to each other.

$\small x=2$

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Question

Which value for satisfies the equation ?

Answer

is the only choice from those given that satisfies the equation. Substition of for gives:

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Question

Solve for (nearest hundredth):

Answer

Take the common logarithm of both sides and solve for :

$\log $8^{y}$ = \log 100$

$y \log 8 =2$

$y =\frac{2}{\log 8}$ \approx $\frac{2}{0.9031}$ \approx 2.21$

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Question

Solve for (nearest hundredth):

$4 ^{2x - 7} = $\frac{1}{256}$$

Answer

$$\frac{1}{256}$ = $\frac{1}{4 ^{4}$} = 4 ^{-4}$ , so $4 ^{2x - 7} = $\frac{1}{256}$$ can be rewritten as

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

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Question

Solve for (nearest hundredth):

$80 ^{t} = 39$

Answer

One method: Take the natural logarithm of both sides and solve for :

$80 ^{t} = 39$

$\ln 80 ^{t} = \ln 39$

$t \ln 80 = \ln 39$

$t =\frac{ \ln 39}{ \ln 80}$ \approx $\frac{ 3.66}{ 4.38}$ \approx 0.84$

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Question

Solve for :

$3 ^{4x + 7} = 9 ^{2x +3}$

Answer

Since $9 = $3^{2}$$ , we can rewrite this equation by subsituting and applying the power rule:

$3 ^{4x + 7} = 9 ^{2x +3}$

$3 ^{4x + 7} = \left (3 ^{2} \right ) ^{2x +3}$

$3 ^{4x + 7} =3 ^{2 \left ( 2x +3 \right )}$

$3 ^{4x + 7} =3 ^{4x +6}$

This statement is identically false, which means that the original equation is identically false. There is no solution.

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Question

Solve for :

$\left ($\frac{1}{4}$ \right) ^{x - 7} = $2^{2x +5}$$

Answer

$\frac{1}{4}$ = $$\frac{1}{2^{2}$$} $=2^{-2}$ , so we can rewrite the equation as follows:

$\left ($\frac{1}{4}$ \right) ^{x - 7} = $2^{2x +5}$$

$\left $(2^{-2}$) ^{x - 7} = $2^{2x +5}$$

$\log_2 $2^{-2x +14 \right)}$ =\log_2 $2^{2x +5}$$

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Question

What are the x-intercepts of the equation?

Answer

To find the x-intercepts, we set the numerator equal to zero and solve.

$$\sqrt{\frac{5}{3}$}=x$

However, the square root of a number can be both positive and negative.

Therefore the roots will be $x=$\sqrt{\frac{5}{3}$}, -$\sqrt{\frac{5}{3}$}.$

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Question

What are the y-intercepts of the equation?

Answer

To find the y-intercepts, set and solve.

$y=\frac{-5}{7}$$

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Question

What are the x-intercepts of the equation?

Answer

To find the x-intercepts, set the numerator equal to zero and solve.

We can simplify from here:

Now we need to rationalize. Because we have a square root on the bottom, we need to get rid of it. Since , we can multiply $$\frac{1}{\sqrt2}$$\frac{\sqrt{2}$}{$\sqrt{2}$}$ to get rid of the radical in the denominator.

$$\frac{1}{\sqrt2}$$\frac{\sqrt{2}$}{$\sqrt{2}$}=x$

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

Since we took a square root, remember that our answer can be either positive or negative, as a positive squared is positive and a negative squared is also positive.

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Question

What are the y-intercepts of the equation?

Answer

To find the y-intercepts, set and solve.

$y=\frac{-2}{-27}$$

$y=\frac{2}{27}$$

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Question

What are the y-intercepts of this equation?

Answer

To find the y-intercept, set and solve.

$y=\frac{-3}{-12}$$

$y=\frac{1}{4}$$

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Question

What are the x-intercepts of the equation?

Answer

To find the x-intercepts, set the numerator equal to zero.

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Question

What are the y-intercepts of this equation?

Answer

To find the y-intercept, set and solve.

$y=\frac{-64}{2}$$

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Question

The population of a certain bacteria increases exponentially according to the following equation:

where P represents the total population and t represents time in minutes.

How many minutes does it take for the bacteria's population to reach 48,000?

Answer

The question gives us P (48,000) and asks us to find t (time). We can substitute for P and start to solve for t:

$48,000 = $2000e^{2t}$$

$24 = $e^{2t}$$

Now we have to isolate t by taking the natural log of both sides:

$\ln 24 = \ln $e^{2t}$$

$\ln 24 = (2t)\ln e$

And since $\ln e = 1$ , t can easily be isolated:

$\ln 24 = 2t$

$$\frac{\ln24}{2}$ = t$

Note: $\frac{\ln 24}{2}$ does not equal $\ln 12$ . You have to perform the log operation first before dividing.

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Question

Solve for .

Answer

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents.

With the same base, we can now write

Subtract on both sides.

Divide on both sides.

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Question

Solve for :

Answer

Because both sides of the equation have the same base, set the terms equal to each other.

Add 9 to both sides:

Then, subtract 2x from both sides:

Finally, divide both sides by 3:

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Question

Solve the equation for .

$\small $9^x$$=3^6$$

Answer

Begin by recognizing that both sides of the equation have a root term of .

$\small $9^x$$=3^6$$

Using the power rule, we can set the exponents equal to each other.

$\small 2x=6$

$\small x=3$

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