Understanding Logarithms - Algebra 2

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Question

What are the domain and the range of the function ?

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Answer

Remember that is still a logarithm of a positive number, .

It's not possible to raise to ANY power and obtain a negative number. Because even $e^{-3}$ , for example, is just $\frac{1}{e^3}$ , which is a ratio of two positive numbers, and therefore positive.

More than that, it's also not possible to obtain 0 by raising to any power. Think: "To what power can I exponentiate e and obtain 0?"

So the domain is strictly positive. It excludes negative numbers and 0.

What about the range? To what possible values are we allowed to exponentiate ?

Well, we just saw that has a definition for negative numbers. (this fact is true for ALL numbers, not just ).

And we can obviously raise it to positive powers. So the range is all real numbers. It includes negative numbers, 0, and positive numbers.

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Question

Based on the definition of logarithms, what is ${\log_{10}1000}$ ?

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Answer

For any equation $\log_{10}(y) = x$ , $10^{x } = y$ . Thus, we are trying to determine what power of 10 is 1000. , so our answer is 3.

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Question

Solve for in the equation:

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Answer

This question tests your understanding of log functions.

can be converted to the form $x=10^{y}$ .

In this problem, make sure to divide both sides by in order to put it in the above form, where . Remember $log(x)=log_{10}(x)$ .

Therefore,

$log_{10}(x)=-4.2 \rightarrow x=10^{-4.2}$

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Question

Which of the following expressions is equivalent to the expression $\log \left ( x^{2} -2x - 8\right )$ ?

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Answer

By the reverse-FOIL method, we factor the polynomial as follows:

$x^{2} -2x - 8 = (x - 4)(x + 2)$

Therefore, we can use the property

$\ln ab = \ln a + \ln b$

as follows:

$\log \left ( x^{2} -2x - 8\right ) = \log [ (x - 4)(x + 2)] = \log (x-4) + \log (x + 2)$

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Question

$7 ^{x} = 1,000$

Evaluate .

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Answer

Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:

$7 ^{x} = 1,000$

$7 ^{x} = 10^{3}$

$\log 7 ^{x} = \log 10^{3}$

$x \log 7 = 3 \log 10$

$x \log 7 = 3 \cdot 1$

$x \log 7 = 3$

$\frac{x \log 7}{\log 7} = \frac{3}{\log 7}$

$x = \frac{3}{\log 7}$

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Question

Evaluate .

$5 ^{x} = 12$

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Answer

Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:

$5 ^{x} = 12$

$\log 5 ^{x} = \log 12$

$x \log 5 = \log 12$

$\frac{x \log 5 }{\log 5 }=\frac{ \log 12}{\log 5}$

$x=\frac{ \log 12}{\log 5}$

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Question

Many textbooks use the following convention for logarithms:

$log(x) = log_{10}(x)$

What is the value of ?

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Answer

Remember:

is the same as saying .

So when we ask "What is the value of ?", all we're asking is "10 raised to which power equals 1,000?" Or, in an expression:

.

From this, it should be easy to see that .

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Question

What is the value of $\log{100000}$ ?

Round to the nearest hundreth.

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Answer

Base-10 logarithms are very easy if the operands are a power of . Begin by rewriting the question:

$\log{100000}$

Becomes...

$\log{10^5}$

because

Applying logarithm rules, you can factor out the :

$5\log{10}$

Now, $\log{10}$ is .

Therefore, your answer is .

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Question

What is the value of $\log{10000000}$ ?

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Answer

Base-10 logarithms are very easy if the operands are a power of . Begin by rewriting the question:

$\log{10000000}$

Becomes...

$\log{10^7}$

because

Applying logarithm rules, you can factor out the :

$7\log{10}$

Now, $\log{10}$ is .

Therefore, your answer is .

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Question

Evaluate the following expression:

$\small \log (1000)$

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Answer

Without a subscript a logarithmic expression is base 10.

The expression $\small \log(1000)=\log _{10}(1000)$

The logarithmic expression is asking 10 raised to what power equals 1000 or what is x when

$\small 10^{x}=1000$

We know that $\small 10^{3}=1000$

so $\small \log (1000)=3$

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Question

Assuming the value of is positive, simplify: $\log_{10}10^{3x+2}$

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Answer

Rewrite the logarithm in division.

$\log_{10}10^{3x+2} = \frac{\log{10}^{3x+2}}{\log 10}$

As a log property, we can pull down the exponent of the power in front as the coefficient.

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

Cancel out the $\log(10)$ .

The answer is:

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Question

Solve the following:

$\log10000+\log10$

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Answer

When the base isn't explicitly defined, the log is base 10. For our problem, the first term

$\log 10000=x$

is asking:

For the second term,

$\log(10)=y$

is asking:

So, our final answer is

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Question

Evaluate .

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Answer

The first thing we can do is bring the exponent out of the log, to the front:

Next, we evaluate :

Recall that log without a specified base is base 10 thus

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

$\log_{10}100=c \Rightarrow 10^c=100 \Rightarrow c=2$ .

Therefore

becomes,

$3\cdot2$ .

Finally, we do the simple multiplication:

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Question

Simplify $log_{10}100$

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Answer

One of the properties of logs is the ability to cancel out terms based on the base of the log. Since the base of the log is 10 we can simplify the 100 to 10 squared.

$log_{10}100=log_{10}10^2$

The log base 10 and the 10 cancel out, leaving you with the value of the exponent, 2 as the answer.

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Question

Evaluate: $log_{10}(2x) =-2$

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Answer

In order to eliminate the log based ten, we will need to raise both sides as the exponents using the base of ten.

The equation becomes:

$10^{log_{10}(2x)} =10^{-2}$

The ten and log based ten will cancel, leaving just the power on the left side. Change the negative exponent into a fraction on the right side.

$2x=\frac{1}{100}$

Divide by two on both sides, which is similar to multiplying by a half on both sides.

$2x\cdot \frac{1}{2}=\frac{1}{100}\cdot \frac{1}{2}$

Simplify both sides.

The answer is: $x=\frac{1}{200}$

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Question

Solve:

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Answer

Break up using log rules. The log has a default base of ten.

$log(A\times B) = log(A)+log(B)$

$log(300) = log(3)+log(100)=log_{10}(3)+log_{10}(10^2)$

The exponent can be brought down as the coefficient since the bases of the second term are common.

$log_{10}(3)+log_{10}(10^2) =log_{10}(3)+2log_{10}(10) = log(3)+2$

This means that:

The answer is:

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Question

What is the value of $log(100^{100})$ ?

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Answer

In order to solve this, we will need to rewrite the base as , since log is by default base 10.

Rewrite the expression.

$log_{10}(10^{2(100)})=log_{10}(10^{200})$

By log rules, the exponent can be pulled down as the coefficient.

$200log_{10}(10) = 200(1) = 200$

The answer is:

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Question

Evaluate: $2log(1000^{30})$

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Answer

The log term has a default base of 10. The 1000 will need to be rewritten as base 10.

$2log(1000^{30}) =2 log_{10}(10^{3(30)})$

Raise the coefficient of the log term as the power.

$log_{10}(10^{3(30)(2)}) = log_{10}(10^{180})$

According to the log property:

$log_{b}(b^x) = x$

The log based 10 and the 10 inside the quantity of the log will cancel, leaving just the power.

The answer is:

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Question

Simplify: $log(1000^{500})$

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Answer

The log has a default of base ten. This means we should convert the 1000 to a common base 10.

Replace this value inside the log term.

$log(1000^{500}) = log_{10}(10^{3(500)})$

Since the log base 10 and the ten to a certain power are existent, they will both cancel, leaving just the power itself.

$log_{b}(b^x) = x$

The answer is:

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Question

Solve: $2log_{10}(100^3)$

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Answer

Change the base of the inner term or log to base ten.

$2log_{10}(100^3)= 2log_{10}(10^{2(3)})= 2log_{10}(10^6)$

According to the log property:

$log_{b}(b^x) = x$

The log based ten and the ten to the power of will cancel, leaving just the power.

$2log_{10}(10^6) = 2(6) =12$

The answer is:

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