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Algebra II : Solving and Graphing Logarithms

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Example Questions

Example Question #21 : Solving Logarithms

Solve the equation

$\ln(3x-2)=\ln(6x)$

Possible Answers:

x=1

No Solution

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

x=2

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

Correct answer:

No Solution

Explanation:

Set the arguments of the natural log equal and solve for

3x-2=6x

-2=3x

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

However, you cannot take the natural log of a negative number. Therefore; there is No Solution

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Example Question #22 : Solving Logarithms

Solve:

$\log_{2}24$

Possible Answers:

$\log24$

$\log12$

$\frac{\log24}{\log2}$

$-\frac{\log24}{\log2}$

Correct answer:

$\frac{\log24}{\log2}$

Explanation:

To solve this problem, it is easiest to start by thinking about what the logarithm means:

2^x=24

Now, we take the log of both sides (either base-10 or natural):

$\log(2^x)=\log24$

Next, using the properties of logarithms, we can bring the exponent x in front of the logarithm:

$x\log2=\log24$

Finally, solve for x:

$x=\frac{\log24}{\log2}$

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Example Question #23 : Solving Logarithms

Evaluate

$\log_5 25 = x$

Possible Answers:

Correct answer:

Explanation:

Logarithms are inverses of exponents.

$\log_5 25$ is asking how many fives multiplied together are equal to 25.

The formula to solve logarithmic equations is as follows.

$log_ab=c\Rightarrow a^c=b$

Applying this formula to our particular problem results in the following.

$\log_5 25\Rightarrow 5^x=25$

5^2 = 5*5 = 25

So x= 2 .

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Example Question #161 : Logarithms

Solve:

$\log_84+\log_810$

Possible Answers:

$\frac{\log40}{\log8}$

$\log5$

$\frac{\log14}{\log8}$

$\log32$

Correct answer:

$\frac{\log40}{\log8}$

Explanation:

To solve, we must keep in mind that a logarithm is asking the following:

$\log_ab$ means a^x=b , and we must find x.

First, we can rewrite the sum of the base-8 logarithms as a product (this is a property of logarithms):

$\log_8(4\cdot 10)=\log_840$

Now, we use the above formula to find what this numerically equals:

8^x=40

Take the natural or common logarithm of both sides, which allows us to bring the power x in front of the logarithm:

$\log8^x=\log40$

$x\log8=\log40$

$x=\frac{\log40}{\log8}$

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Example Question #25 : Solving Logarithms

Solve for x:

$4^x=3^{x^2}$

Possible Answers:

$\log\frac{3}{4}$

$\frac{\log4}{\log3}$

$\frac{\log3}{\log4}$

$\log1$

$\log\frac{4}{3}$

Correct answer:

$\frac{\log4}{\log3}$

Explanation:

To solve for x, we must take the logarithm of both sides (common or natural, it doesn't matter):

$\log(4^x)=\log(3^{x^2})$

In doing this, we can now bring the exponents in front of the logarithms:

$x\log4=x^2\cdot\log3$

Now, solve for x:

$\log4=x\log3$

$x=\frac{\log4}{\log3}$

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Example Question #161 : Logarithms

If $log_{m}(2)=.387$ and $log_{m}(3)=.613$ , what is ?

Possible Answers:

Correct answer:

Explanation:

First, we know from logarithmic properties that:

$log_{n}(n)=1$

So if we can find a combination of our knowns that equal , we should be able to figure out . It may take some guess and check, but:

.387 + .613 = 1

That would equate to:

$log_{m}(2)+log_{m}(3) = 1$

Using our log properties, we can simplify into:

$log_{m}(2\cdot 3)=1$

$log_{m}(6)=1$

That means that:

m=6

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Example Question #27 : Solving Logarithms

Evaluate $log_{100}(10000)$ .

Possible Answers:

100

Correct answer:

Explanation:

There are a few ways to go about this, but let's use a change of base to make the problem easier to work without using a calculator. First, we know that:

$log_{m}(n)=\frac{log_x(m)}{log_x(n)}$

We can choose any value for that we would like here, so to make things simple, let's put this problem in base :

$log_{100}(10000) = \frac{log_{10}(10000)}{log_{10}(100)} = \frac{4}{2} = 2$

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Example Question #28 : Solving Logarithms

Evaluate log_3(27) .

Possible Answers:

Correct answer:

Explanation:

We can take this problem and expand it a bit, which will make things easier in the long run. We know that:

$log_3(27)=log_3(3\cdot 3\cdot 3)$

Using one of our logarithmic properties, we can expand even further:

$log_3(3\cdot 3\cdot 3)=log_3(3)+log_3(3)+log_3(3)$

Another log property states that:

log_x(x)=1

So:

log_3(3)+log_3(3)+log_3(3) = 1+1+1 = 3

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Example Question #21 : Solving And Graphing Logarithms

Solve:

$\log_4 16^x=10$

Possible Answers:

$x=\frac{5}{2}$

x=5

x=1

x=0

$x=\frac{2}{5}$

Correct answer:

x=5

Explanation:

To solve for x, remember that exponents inside logarithms can be moved to the front of the logarithm:

$x\cdot \log_416=10$

Next, we can rewrite the logarithm as the number it equals, and solve for x:

$x\cdot2=10$

x=5

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Example Question #22 : Solving And Graphing Logarithms

Solve: ln(2x) = 100

Possible Answers:

$e^{50}$

$\frac{e^{100}}{2}$

$2e^{100}$

Correct answer:

$\frac{e^{100}}{2}$

Explanation:

In order to solve for the x-variable, we will need to exponential function both sides of the equation in order to eliminate the natural log.

$e^{ln(2x) }= e^{100}$

The equation becomes:

$2x= e^{100}$

Divide by two on both sides.

$\frac{2x}{2}= \frac{e^{100}}{2}$

$x= \frac{e^{100}}{2}$

The answer is: $\frac{e^{100}}{2}$

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Algebra II : Solving and Graphing Logarithms

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Solving Logarithms

Example Question #22 : Solving Logarithms

Example Question #23 : Solving Logarithms

Example Question #161 : Logarithms

Example Question #25 : Solving Logarithms

Example Question #161 : Logarithms

Example Question #27 : Solving Logarithms

Example Question #28 : Solving Logarithms

Example Question #21 : Solving And Graphing Logarithms

Example Question #22 : Solving And Graphing Logarithms

All Algebra II Resources

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