Solve the exponential equation 3^x – 5 = 15y + 5 for x.

First, isolate the term with x in it:

3^x = 15y + 10

Then take the base-3 logarithm of both sides:

log₃(3^x) = log₃(15y + 10)

Simplify, using the rules of logarithms:

x = log₃(15y + 10)

Bam. Knocked it outta the park.

Change log₄ 20 to base-10 and solve to four decimal places.

Popping our values into the change of base formula, we get:

Punch both into our calculator and we'll get a numerator of about 1.3010 and a denominator of 0.6021. Divide 'em to get our final answer: 2.1608.

Find the inverse of y = (7^x + 5)^1/5.

First we raise both sides to the 5th power to get rid of that awful fractional exponent:

y⁵ = 7^x + 5

Subtract 5 from both sides:

y⁵ – 5 = 7^x

Take the base-7 log of both sides to yank that x out of the exponent:

log₇(y⁵ – 5) = log₇(7^x)

The right side cancels:

log₇(y⁵ – 5) = x

Switch x and y:

y = log₇(x⁵ – 5)

And there's our inverse. It was like a walk in the park, if the park were filled with vicious exponents and logarithms.

Show to three decimal places that log₄ 64 = log₇ 343 using base-10 logarithms.

We just need to show that the two sides are equal. First, use the change of base formula on both sides:

Solve each individually:

Simplify:

3 = 3

True!