Algebra II - Logarithms and exponents

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 .

True

False

Explanation

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:

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

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

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

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

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

Try to answer without a calculator.

True or false:

$\log_{7} 700 = 2$

False

True

Explanation

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.

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

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

$\log_{N }100 = 2M$

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

$\log_{N }100 = M + 2$

$\log_{N }100 = M + 1$

$\log_{N }100 = \frac{M}{2}$

Explanation

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 .

Try without a calculator:

Evaluate $\log_{16} 8$

$\frac{3}{4}$

$\frac{4}{3}$

$-\frac{4}{3}$

$-\frac{3}{4}$

None of the other choices gives the correct response.

Explanation

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

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

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

It follows that

and

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

the correct response.

Solve for :

Round to the nearest hundredth.

Cannot be computed

Explanation

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,

Simplify:

Explanation

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:

Solve for :

Round to the nearest hundredth.

Explanation

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: