Chapters

Logarithms by Definition
Logarithm Calculation
Logarithm Development

To solve these exercises, let's remember the definition of a logarithm which tells us that if $\text{[math]}$ is equal to the logarithm, base $\text{[math]}$ , of $\text{[math]}$

$\text{[math]}$

implies that $\text{[math]}$ .

Now that we have that in mind, let's proceed to solve the exercises.

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Logarithms by Definition

Applying the definition of logarithm, calculate the value of y

Score:

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

We apply the definition of logarithm and convert $\text{[math]}$ to a fraction, that is $\text{[math]}$ , then simplify

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

We apply the definition of logarithm

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

Note that when we write $\text{[math]}$ we refer to base $\text{[math]}$ , that is $\text{[math]}$

We apply the definition of logarithm

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

Remember that the natural logarithm is simply the logarithm base $\text{[math]}$ , that is, $\text{[math]}$ . We apply the definition of logarithm and solve

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

We apply the definition of logarithm and solve

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

We apply the definition of logarithm and solve

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

We apply the definition of logarithm and solve

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

We apply the definition of logarithm and solve

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

We apply the definition of logarithm and solve.

Note that in this case it is a bit different since $\text{[math]}$ is the base of the logarithm.

$\text{[math]}$

Solution

Our expression is

$\text{[math]}$

We apply the definition of logarithm and solve. Note that in this case it is a bit different since $\text{[math]}$ is found in the argument of the logarithm.

$\text{[math]}$

Logarithm Calculation

In these exercises we will apply the change of base property of logarithms, which tells us that the logarithm, base $\text{[math]}$ , of $\text{[math]}$ is equal to

$\text{[math]}$

for another base $\text{[math]}$ . Note that the expression on the right is already in new base $\text{[math]}$ .

Score:

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

Let's proceed by converting the argument to an appropriate fraction

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

We proceed by writing $\text{[math]}$ as a power of $\text{[math]}$ .

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

We proceed by writing $\text{[math]}$ as $\text{[math]}$ and then apply some properties of logarithms

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

We proceed by writing $\text{[math]}$ as $\text{[math]}$ and then apply some properties of logarithms

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

We proceed by writing $\text{[math]}$ as a fraction in which there is a power of $\text{[math]}$ and apply properties of logarithms

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

Let's proceed by converting the argument to an appropriate fraction

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

We proceed by writing $\text{[math]}$ as a power of $\text{[math]}$ .

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

We proceed by writing $\text{[math]}$ as $\text{[math]}$ and then apply some properties of logarithms

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

We proceed by writing $\text{[math]}$ as a fraction in which there is a power of $\text{[math]}$ and apply properties of logarithms

$\text{[math]}$

Given $\text{[math]}$ , calculate the following logarithm:

$\text{[math]}$

Solution

Our expression to solve is

$\text{[math]}$

We proceed by writing $\text{[math]}$ as a power of $\text{[math]}$ .

$\text{[math]}$

Logarithm Development

Develop the following expressions

Score:

$\text{[math]}$