Converting exponentials into logarithms

We know how to solve and manipulate logarithms (see laws of logarithms), but what if we have an exponential expression that we want to convert into a logarithmic one?

We can just use the definition of logarithms to rewrite the exponential expression a^x=b as a logarithmic expression x=\log_a(b).

Convert 2^3=8 into a logarithmic expression

• \log_2(8)

Convert 5^x=25 into a logarithmic expression

• x=\log_5(25)
• x=2

Solve for x: 3^x=27

• x=\log_3(27)
• x=3

Solve for x: 10^{x-1}=1000

• x-1=\log_{10}(1000)
• x-1=3
• x=4

Find the exact value of x: e^{2x}=7

• 2x=\ln(7)
• x=\frac{\ln(7)}{2}

Find the exact value of x: 4^{x+1}=16

• x+1=\log_4(16)
• x+1=2
• x=1

Solve for x: (2^x)^3=64

• 3x=\log_2(64)
• 3x=6
• x=2

Solve for x: (5^{2x-1})^2=125

• 5^{2(2x-1)}=125
• 5^{4x-2}=125
• 4x-2=\log_5(125)
• 4x-2=3
• 4x=5
• x=\frac{5}{4}

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