Powers of logarithms

If we see a logarithm in a power, and the base of the logarithm is the same as the base of the power, we know that:

a^{\log_a{x}}=x

This is very similar (but just the other way round) to the self-base logarithms topic.

Evaluate 2^{\log_2{5}}

• a^{\log_a{x}}=x
• 2^{\log_2{5}}=5
• Answer: 5

Evaluate 10^{\log_{10}{0.01}}

• a^{\log_a{x}}=x
• 10^{\log_{10}{0.01}}=0.01
• Answer: 0.01

Evaluate 3^{\log_3{9}}

• a^{\log_a{x}}=x
• 3^{\log_3{9}}=9
• Answer: 9

Evaluate 5^{\log_5{25}}

• a^{\log_a{x}}=x
• 5^{\log_5{25}}=25
• Answer: 25

Evaluate 7^{\log_7{6^4}}

• a^{\log_a{x}}=x
• 7^{\log_7{6^4}}=6^4
• 6^4=1296
• Answer: 1296

Simplify 4^{\log_4{(2x+1)}}

• a^{\log_a{y}}=y
• 4^{\log_4{(2x+1)}}=2x+1
• Answer: 2x+1

Simplify 8^{\log_8{(x^2+1)}}

• a^{\log_a{y}}=y
• 8^{\log_8{(x^2+1)}}=x^2+1
• Answer: x^2+1

Simplify k^{\log_k{(x^3+2)}}

• a^{\log_a{y}}=y
• k^{\log_k{(x^3+2)}}=x^3+2
• Answer: x^3+2

Given that x^{\log_x{y}}=16, find the value of y

• a^{\log_a{y}}=y
• x^{\log_x{y}}=y
• y=16
• Answer: 16

Given that m^{\log_m{n}}=64, find the value of n

• a^{\log_a{n}}=n
• m^{\log_m{n}}=n
• n=64
• Answer: 64