Solve 12^x = 14^3x.

I know I need to make the bases the same, but I don't know how.

just take log of both sides

x log12 = 3x log14
x(log12 - 3log14) = 0
x=0

the two exponential functions both pass through (0,1), but that is the only intersection.

To solve the equation 12^x = 14^3x, you're correct in wanting to make the bases the same.

To do this, you can take the logarithm of both sides of the equation. Logarithms allow you to work with exponentials in a more simplified way.

In this case, you can use either the natural logarithm (ln) or the common logarithm (log base 10) to find the solution. Let's use the natural logarithm (ln) for this example.

Taking the natural logarithm of both sides of the equation gives:
ln(12^x) = ln(14^3x).

Using the power rule of logarithms, you can bring down the exponent in front:
x * ln(12) = 3x * ln(14).

Now, you can divide both sides of the equation by x to isolate the terms:
ln(12) = 3ln(14).

Finally, divide both sides by ln(14) to solve for x:
ln(12) / ln(14) = 3.

Using a calculator to compute the left side of the equation, you can find the value of ln(12) / ln(14). The result is approximately 1.174.

Therefore, the solution to the equation 12^x = 14^3x is x ≈ 1.174.