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Algebra II : Natural Log

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Example Questions

Example Question #21 : Natural Log

Solve: $\displaystyle ln(2x-e) = 3$

Possible Answers:

$\frac{e^3+e}{2}$ $\displaystyle \frac{e^3+e}{2}$

$\frac{(e+1)^3}{2}$ $\displaystyle \frac{(e+1)^3}{2}$

$\frac{e^3-e}{2}$ $\displaystyle \frac{e^3-e}{2}$

$\frac{3}{2}e$ $\displaystyle \frac{3}{2}e$

$\frac{9-e}{3}$ $\displaystyle \frac{9-e}{3}$

Correct answer:

$\frac{e^3+e}{2}$ $\displaystyle \frac{e^3+e}{2}$

Explanation:

In order to eliminate the natural log, which has a base of $\displaystyle e$ , we will need to raise both side as powers of $\displaystyle e$ .

$e^{ln(2x-e)} = e^3$ $\displaystyle e^{ln(2x-e)} = e^3$

The equation can be simplified to:

$\displaystyle 2x-e = e^3$

Add $\displaystyle e$ on both sides.

$\displaystyle 2x-e +e= e^3+e$

$\displaystyle 2x= e^3+e$

Divide by two on both sides.

$\frac{2x}{2}= \frac{e^3+e}{2}$ $\displaystyle \frac{2x}{2}= \frac{e^3+e}{2}$

The answer is: $\frac{e^3+e}{2}$ $\displaystyle \frac{e^3+e}{2}$

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Example Question #261 : Mathematical Relationships And Basic Graphs

Try without a calculator:

Which expression is not equivalent to 1?

Possible Answers:

$1,000,000^{0}$ $\displaystyle 1,000,000^{0}$

$\log 10$ $\displaystyle \log 10$

$i^{4}$ $\displaystyle i^{4}$

$1,000^{\frac{1}{1,000}}$ $\displaystyle 1,000^{\frac{1}{1,000}}$

$\ln e$ $\displaystyle \ln e$

Correct answer:

$1,000^{\frac{1}{1,000}}$ $\displaystyle 1,000^{\frac{1}{1,000}}$

Explanation:

$1,000^{\frac{1}{1,000}}$ $\displaystyle 1,000^{\frac{1}{1,000}}$ is the correct choice.

For all a,N $\displaystyle a,N$ for which the expressions are defined,

$a ^{\frac{1}{N}} = \sqrt[N]{a}$ $\displaystyle a ^{\frac{1}{N}} = \sqrt[N]{a}$ .

Setting $\displaystyle a=1,000, N = 1,000$ , this equation becomes

$1,000 ^{\frac{1}{1,000}} = \sqrt[1,000]{1,000}$ $\displaystyle 1,000 ^{\frac{1}{1,000}} = \sqrt[1,000]{1,000}$ - that is, the one thousandth root of 1,000. This is not equal to 1, since if it were, it would hold that $1^{1,000} = 1,000$ $\displaystyle 1^{1,000} = 1,000$ - which is not true.

Of the other four expressions:

$\log 10$ $\displaystyle \log 10$ , the common, or base ten, logarithm of 10, can be rewritten as $\log_{10}10$ $\displaystyle \log_{10}10$ , and $\ln e$ $\displaystyle \ln e$ , the natural, or base $\displaystyle e$ , logarithm of $\displaystyle e$ , can be rewritten as $\log_{e}e$ $\displaystyle \log_{e}e$ . A property of logarithms states that for all $N > 0, N \ne 1$ $\displaystyle N > 0, N \ne 1$ , $\log_{N}N = 1$ $\displaystyle \log_{N}N = 1$ . Therefore, $\log 10 = \log_{10}10 = 1$ $\displaystyle \log 10 = \log_{10}10 = 1$ and $\ln e = \log_{e}e = 1$ $\displaystyle \ln e = \log_{e}e = 1$ .

$1,000,000^{0} = 1$ $\displaystyle 1,000,000^{0} = 1$ , since any nonzero number raised to the power of 0 is equal to 1.

By the Power of a Power Property,

$i^{4} = i ^{2 \cdot 2} =( i^{2} )^{2}$ $\displaystyle i^{4} = i ^{2 \cdot 2} =( i^{2} )^{2}$

$i^{2} = -1$ $\displaystyle i^{2} = -1$ , so

$i^{4} = ( -1)^{2}= 1$ $\displaystyle i^{4} = ( -1)^{2}= 1$

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Algebra II : Natural Log

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Natural Log

Example Question #261 : Mathematical Relationships And Basic Graphs

All Algebra II Resources

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