To solve this problem, recall that you can set exponents equal to eachother if they have the same base.

See below:

So, we have

Because both sides of this equation have a base of seven, we can set the exponents equal to eachother and solve for t.

We begin by taking the natural log of the equation:

Simplifying the left side of the equation using the rules of logarithms gives:

We group the x terms to get:

We reincorporate the exponents into the logarithms and use the identity property of the natural log to obtain:

我们结合logarithms using the multiplication/sum rule to get:

We then solve for x:

We begin by factoring out the termto get:

This equation gives our first solution:

Then we check for more solutions:

Therefore our solution is

Remember the denominator of a rational exponent is equivalent to the index of a root.

This should simplify quite nicely.

Whenit gives us,

What does an exponent of one-third mean? Consider our expression and raise it to the third power.

Simplifying, we get:

Thus, we are looking for a number that when cubed, we get. Thus, we are discussing the cube root of, or.

The derifintion of logarithm is:

In this problem,

Therefore,

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'.

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 solve the equation as follows:

Exponentiate both sides.

Apply the power rule on the right hand side.

Multiply by.

Divide by.

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 byon both sides:

take the cube root: