Logarithms and exponents - Algebra II

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Question

What is the value of that satisfies the equation $\log_x128=7$ ?

Answer

$\log_xa=b$ is equivalent to . In this case, you know the value of (the argument of a logarithmic equation) and b (the answer to the logarithmic equation). You must find a solution for the base.

$x^{7}=128$

$\sqrt[7]{x}=\sqrt[7]{128}$

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Question

Which equation is equivalent to:

$\log _{4}\left( \frac{1}{2}\right)=x$

Answer

$log{_{a}}^{b}=x$ ,

$\Rightarrow a^{x}=b$

So, ${{log_{4}}^{\frac{1}{2}}}=x\Rightarrow 4^{x}=\frac{1}{2}$

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Question

$Log_{b}x=y$

What is the inverse of the log function?

Answer

This is a general formula that you should memorize. The inverse of $Log_{b}x=y$ is $b^{y}=x$ . You can use this formula to change an equation from a log function to an exponential function.

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Question

Solve for :

Round to the nearest hundredth.

Answer

To solve this, you need to set up a logarithm. Our exponent is . The logarithm's base is . The value is the operand of the logarithm. Therefore, we can write an equation:

$\log_7{54} = b$

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

$\log_7{54}$

to...

$\frac{\log{54}}{\log{7}}$

You can put this into your calculator and get:

, or rounded,

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Question

Solve for :

Round to the nearest hundredth.

Answer

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of is the base. This is the logarithm's base. The value is the operand of the logarithm. Therefore, we can write an equation:

$\log_3_1{842} = r$

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

$\log_3_1{842}$

to...

$\frac{\log{842}}{\log{31}}$

You can put this into your calculator and get:

, or rounded,

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Question

Solve for :

Round to the nearest hundredth.

Answer

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of is the base. This is the logarithm's base. The value is the operand of the logarithm. Therefore, we can write an equation:

$\log_4 {6317} = x$

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

$\log_4 {6317}$

to...

$\frac{\log{6317}}{\log{4}}$

You can put this into your calculator and get:

, or rounded,

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Question

Solve for :

Round to the nearest hundredth.

Answer

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of is the base. This is the logarithm's base. The value is the operand of the logarithm. Therefore, we can write an equation:

$\log_5 {40150} = t$

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

$\log_5 {40150}$

to...

$\frac{\log{40150}}{\log{5}}$

You can put this into your calculator and get:

, or rounded,

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Question

Write the equation $3^{-4}=\frac{1}{81}$ in logarithmic form.

Answer

For logarithmic equations, can be rewritten as .

In this expression, is the base of the equation (). is the exponent () and $\frac{1}{81}$ is the term ().

In putting each term in its appropriate spot, the exponential equation can be converted to $log_3\frac{1}{81}=-4$ .

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Question

Solve the following logarithm for :

Answer

Solve the following logarithm:

Recall that we can convert logarithms to exponential form via the following:

Using this approach, convert the given log to exponential form:

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Question

Rewrite the following expression as an exponential expression:

Answer

Rewrite the following expression as an exponential expression:

Recall the following property of logs and exponents:

Can be rewritten in the following form:

So, taking the log we are given;

We can rewrite it in the form:

$3^{14,000}=b$

So b must be a really huge number!

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Question

Convert the following logarithmic equation to an exponential equation:

$log_{13}b=147$

Answer

Convert the following logarithmic equation to an exponential equation:

$log_{13}b=147$

Recall the following:

This

Can be rewritten as

So, our given logarithm

$log_{13}b=147$

Can be rewritten as

$13^{147}=b$

Fortunately we don't need to expand, because this woud be a very large number!

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Question

Rewrite the follwing equation as a logarithm:

Answer

Rewrite the follwing equation as a logarithm:

To complete this problem, recall the following relationship:

can be rewritten as

So, this:

Is the same thing as this:

$log_{14}977=b$

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Question

Simplify:

Answer

When the base is raised to a certain power, taking the natural log of this whole term will eliminate the exponential and the power can be pulled out as the coefficient.

The answer is:

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Question

Solve: $log_{\frac{1}{5}} 125$

Answer

In order to solve this log, we will need to write 125 in terms of one fifth to a certain power.

Rewrite 125 as an exponent of one-fifth.

$125 = (\frac{1}{5})^{-3}$

$log_{\frac{1}{5}} 125=log_{\frac{1}{5}}(\frac{1}{5})^{-3}$

According to the log rule, $log_{x}x^y=y$ , the bases will cancel, leaving just the exponent.

The answer is:

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Question

Simplify, if possible:

Answer

Notice that the term in the log can be rewritten as a base raised to a certain power.

Rewrite the number in terms of base five.

According to log rules, the exponent can be dropped as a coefficient in front of the log.

$2\cdot 3log(5) = 6log(5)$

The answer is:

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Question

Solve: $3log_{5}(5^{10})$

Answer

Evaluate the log using the following property:

The log based and the base of the term will simplify.

The expression becomes:

$3log_{5}(5^{10}) = 3(10) = 30$

The answer is:

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Question

Given the following:

$\log_{8} M = N$

Decide if the following expression is true or false:

$\log_{M} 8 = \frac{1}{N}$ for all positive .

Answer

By definition of a logarithm,

$\log_{8} M = N$

if and only if

$8^{N } = M$

Take the th root of both sides, or, equivalently, raise both sides to the power of $\frac{1}{N}$ , and apply the Power of a Power Property:

$\left (8^{N } \right ) ^{\frac{1}{N}} = M ^{\frac{1}{N}}$

$8^{N \cdot \frac{1}{N} } = M ^{\frac{1}{N}}$

$8^{1 } = M ^{\frac{1}{N}}$

or

$M ^{\frac{1}{N}} = 8$

By definition, it follows that $\log_{M} 8 = \frac{1}{N}$ , so the statement is true.

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Question

$\log_{N }10 = M$ , with positive and not equal to 1.

Which of the following is true of $\log_{N }100$ for all such ?

Answer

By definition,

$\log_{N }10 = M$

If and only if

$N^{M}= 10$

Square both sides, and apply the Power of a Power Property to the left expression:

$(N^{M})^{2}= 10^{2}$

$N^{M\cdot 2}= 100$

$N^{2M}= 100$

It follows that for all positive not equal to 1,

$\log_{N }100 = 2M$

for all .

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Question

Try to answer without a calculator.

True or false:

$\log_{7} 700 = 2$

Answer

By definition, $\log_{7} 700 = 2$ if and only if $7^{2} = 700$ . However,

$7^{2} = 7 \times 7 = 49 e 700$ ,

making this false.

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Question

Try without a calculator:

Evaluate $\log_{16} 8$

Answer

By definition, $\log_{16} 8 = N$ if and only if $16^{N} = 8$ .

8 and 16 are both powers of 2; specifically, $8= 2^{3} , 16 = 2^{4}$ . The latter equation can be rewritten as

$\left (2 ^{4} \right ) ^{N} = 2^{3}$

By the Power of a Power Property, the equation becomes

$2 ^{4\cdot N} = 2^{3}$

or

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

It follows that

,

and

$N = \frac{3}{4}$ ,

the correct response.

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