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Example Question #1 : Exponential And Logarithmic Functions
You are given that and
.
Which of the following is equal to ?
Since and
, it follows that
and
Example Question #1 : Logarithmic Functions
Solve.
None of the other answers
First we condense the LHS to using the properties of logarithms. Then we can eliminate the natural log on the LHS by raising both sides of the equation as exponents with base
. This leaves us with
since
. Now we do some simple Algebra and obtain our answer.
Example Question #1 : Exponential And Logarithmic Functions
Solve.
None of the other answers
First take the logarithm of both sides. This yields . Now using the properties of logarithms, we can bring the
down in front of
. Since
. We are left with
. Dividing each side by
we find
.
Example Question #4 : Exponential And Logarithmic Functions
Evalute the equation . Use a calculator to approximate the answer to three decimal places.
First, we rewrite the equation in logarithmic form to obtain . Simplifying, we get
Example Question #1 : Logarithmic Functions
Simplify
Example Question #1 : Exponential And Logarithmic Functions
Solve the following equation for :
Logs must be used to solve this problem. Follow the steps below:
Example Question #1 : Exponential And Logarithmic Functions
Solve.
First we condense the LHS to . Now we can eliminate a log of base
by making both sides of the equation exponents of base
. This leaves us with
. Moving
to the LHS we obtain a simple quadratic which we solve to obtain
and
for solutions. Finally, we eliminate
as logarithms are undefined for negative numbers.
Example Question #1 : Logarithmic Functions
Find the domain of the function
.
The logarithmic function is undefined when the inputs are negative or 0. Therefore the inputs of the logarithmic function must be positive. This means that the quantity must be positive. After setting up the appropriate inequality, we have,
Therefore the domain of the function
is the interval
.
Example Question #1 : Exponential And Logarithmic Functions
Use the logarithmic base change formula to convert to a quotient of logarithms with a base of
.
Here we employ the use of the logarithm base change formula
.
Therefore we get,
Example Question #1 : Logarithmic Functions
Find the domain of the function .
The logarithmic function is undefined when the inputs are negative or 0. Therefore the inputs of the logarithmic function must be positive. This means that the quantity must be positive. However, this quantity is always positive because the value under the radical cannot be negative. So we are only constrained by:
Therefore the domain of the function is the interval
.
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