Exponential Functions - College Algebra

Card 0 of 9

Question

Which equation is equivalent to:

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

Answer

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

$\Rightarrow a^{x}=b$

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

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

Question

Solve:

Answer

To solve , it is necessary to know the property of .

Since and the terms cancel due to inverse operations, the answer is what's left of the term.

The answer is:

Compare your answer with the correct one above

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!

Compare your answer with the correct one above

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!

Compare your answer with the correct one above

Question

Convert the following logarithmic equation to an exponential equation.

Answer

Convert the following logarithmic equation to an exponential equation.

To convert from logarithms to exponents, recall the following property:

Can be rewritten as:

So, starting with

,

We can get

$3^{156}=14x$

Compare your answer with the correct one above

Question

Solve the following:

Answer

To solve the following, you must "undo" the 5 with taking log based 5 of both sides. Thus,

$\log_5{5^x}=\log_5{125}$

$x=\log_5{125}$

The right hand side can be simplified further, as 125 is a power of 5. Thus,

$x=\log_5{125}$

$x=\log_5{5^3}$

Compare your answer with the correct one above

Question

Solve for :

$2^{x}+ 2 ^{x+2} = 171$

(Nearest hundredth)

Answer

Apply the Product of Powers Property to rewrite the second expression:

$2^{x}+ 2 ^{x+2} = 171$

$2^{x}+ 2 ^{2} \cdot 2 ^{x} = 171$

$1 \cdot 2^{x}+ 4 \cdot 2 ^{x} = 171$

Distribute out:

$(1 + 4) \cdot 2 ^{x} = 171$

$5 \cdot 2 ^{x} = 171$

Divide both sides by 5:

$\frac{5 \cdot 2 ^{x}}{5} = \frac{171}{5}$

$2 ^{x} = 34.2$

Take the natural logarithm of both sides (and note that you can use common logarithms as well):

$\ln 2 ^{x} = \ln 34.2$

Apply a property of logarithms:

$x \ln 2 = \ln 34.2$

Divide by $\ln 2$ and evaluate:

$\frac{x \ln 2}{\ln2} = \frac{\ln 34.2}{\ln2}$

$x = \frac{3.5322 }{0.6931 }$

$x \approx 5.10$

Compare your answer with the correct one above

Question

Solve for :

$3^{2x+ 1} = 9^{2x- 4 }$

(Nearest hundredth, if applicable).

Answer

$9 = 3^{2}$ , so rewrite the expression at right as a power of 3 using the Power of a Power Property:

$3^{2x+ 1} = 9^{2x- 4 }$

$3^{2x+ 1} = (3^{2} )^{2x- 4 }$

$3^{2x+ 1} = 3^{2(2x- 4 )}$

Set the exponents equal to each other and solve the resulting linear equation:

Distribute:

Subtract and 1 from both sides; we can do this simultaneously:

Divide by :

$\frac{- 2x}{-2} = \frac{-9}{-2}$

Compare your answer with the correct one above

Tap the card to reveal the answer