Logarithms: Worked Examples

The nine problems progress from converting forms to equation solving and interpreting an idealized logarithmic ratio. Every real logarithm requires a base greater than zero and different from one, and a strictly positive argument.

Convert Between Exponential and Logarithmic Forms

Convert 25=322^5=32 to logarithmic form and log⁡3(81)=4\log_3(81)=4 to exponential form.

Step-by-Step Solution

0 of 3 Steps Completed
1

Evaluate Three Basic Logarithms Exactly

Evaluate log⁡4(64)\log_4(64), log⁡5(1/25)\log_5(1/25), and log⁡16(4)\log_{16}(4) without a calculator.

Step-by-Step Solution

0 of 3 Steps Completed
1

Evaluate a Logarithm with a Radical Argument

Calculate log⁡2(163)\log_2(\sqrt[3]{16}) exactly.

Step-by-Step Solution

0 of 3 Steps Completed
1

Expand a Logarithm Using Three Laws

Expand log⁡b ⁣(x3y2z)\log_b\!\left(\dfrac{x^3y^2}{z}\right), assuming b>0b>0, b≠1b\ne1, and x,y,z>0x,y,z>0.

Step-by-Step Solution

0 of 3 Steps Completed
1

Condense a Combination of Natural Logarithms

Condense 2ln⁡x−12ln⁡y+3ln⁡z2\ln x-\tfrac12\ln y+3\ln z, given x,y,z>0x,y,z>0.

Step-by-Step Solution

0 of 3 Steps Completed
1

Change of Base to Four Decimal Places

Evaluate log⁡7(50)\log_7(50) using common logarithms.

Step-by-Step Solution

0 of 3 Steps Completed
1

Solve an Exponential Equation with Logarithms

Solve 5x+2=72x5^{x+2}=7^{2x} for real xx.

Step-by-Step Solution

0 of 3 Steps Completed
1

Reject an Extraneous Logarithmic Root

Solve log⁡2(x)+log⁡2(x−3)=2\log_2(x)+\log_2(x-3)=2.

Step-by-Step Solution

0 of 4 Steps Completed
1

Interpret an Idealized Base-Ten Ratio Index

An illustrative index is defined by L=log⁡10(I/I0)L=\log_{10}(I/I_0) for a dimensionless positive ratio with I0>0I_0>0. If L=6.0L=6.0, find I/I0I/I_0. This is a mathematics example, not a complete model of earthquake magnitude or a measured seismic intensity.

Step-by-Step Solution

0 of 3 Steps Completed
1