Solve Logarithmic Equations - Pre-Calculus

The derifintion of logarithm is:

In this problem,

Therefore,

The natural logarithm and natural exponent are inverses of each other. Taking the of will simply result in the argument of the exponent.

That is

Now, , so

Using the properties of logarithms,

the expression can be rewritten as

which simplifies to .

Using the rules of logarithms,

Hence,

So exponentiate both sides with a base 10:

The exponent and the logarithm cancel out, leaving:

This answer does not match any of the answer choices, therefore the answer is 'None of the other choices'.

You need to know the Laws of Logarithms in order to solve this problem. The ones specifically used in this problem are the following:

Let's take this one variable at a time starting with expanding z:

Now y:

And finally expand x:

By the property of the sum of logarithms,

.

In order to solve this equation, we must apply several properties of logarithms. First we notice the term on the left side of the equation, which we can rewrite using the following property:

Where a is the coefficient of the logarithm and b is some arbitrary base. Next we look at the right side of the equation, which we can rewrite using the following property for the addition of logarithms:

Using both of these properties, we can rewrite the logarithmic equation as follows:

We have the same value for the base of the logarithm on each side, so the equation then simplifies to the following:

Which we can then factor to solve for :

We start expanding our logarithm by using the following property:

Now we have two terms, and we can further expand the first term with the following property:

Now we only have two logarithms left with nonlinear terms, which we can expand using one final property:

Using this property on our two terms with exponents, we obtain the final expanded expression:

We start condensing our expression using the following property, which allows us to express the coefficients of two of our terms as exponents:

Our next step is to use the following property to combine our first three terms:

Finally, we can use the following property regarding subtraction of logarithms to obtain the condensed expression:

The correct answer is

Properties of logarithms allow us to rewrite and as and , respectively. So we have

Again, we use the logarithm property

to get

We solve the equation as follows:

Exponentiate both sides.

Apply the power rule on the right hand side.

Multiply by .

Divide by .

To expand

, use the quotient property of logs.

The quotient property states:

Substituting in our given information we get:

Using the properties of logarithms, expand the logrithm one step at a time:

When expanding logarithms, division becomes subtration, multiplication becomes division, and exponents become coefficients.

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Begin by recalling a few logarithm rules:

1. When adding logarithms of like base, multiply the inside.

2. When subtracting logarithms of like base, subtract the inside.

3. When multiplying a logarithm by some number, raise the inside to that power.

Keep these rules in mind as we work backward to solve this problem:

Using rule 2), we can get the following:

Next, use rule 1) on the first part to get:

Finally, use rule 3) on the second and third parts to get our final answer:

First, simplify the logarithmic expressions on the left side of the equation:

can be re-written as .

Now we have:

.

The left can be consolidated into one log expression using the subtraction rule:

.

We now have log on both sides, so we can be confident that whatever is inside these functions is equal:

to continue solving, multiply by on both sides:

take the cube root:

First bring the inside exponent in front of the natural log.

.

Next simplify the first term and bring all the terms on one side of the equation.

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Next, let set

, so .

Now use the quadratic formula to solve for .

and thus, and .

Now substitute with .

So, since and .

Thus, .

Use the Power property of Logarithms:

Rewrite the fractional exponent:

Condense into a fraction using the Quotient property of Logarithms:

Use the Quotient property of Logarithms to express on a single line:

Use the Product property of Logarithms to expand the two terms further:

Finally use the Power property of Logarithms to remove all exponents:

The expression is now fully expanded.

When logs of the same bases are subtracted, the contents of both logs will be divided with each other. When logs of the same bases are added, then the contents inside the log will be multiplied together.