Adding logarithms

When adding together two logarithms with the same base, it’s equivalent to multiplying the values inside the logarithms:

\log_b(x) + \log_b(y) = \log_b(xy)

Adding two logarithms

Example: Evaluate \log_2(8) + \log_2(4)

• Both have the same base of 2.
• Multiply the values inside: 8 \times 4 = 32.
• So, \log_2(8) + \log_2(4) = \log_2(32).
• \log_2(32)=5 because 2^5=32.
• Answer: 5.

Example: Evaluate \log_{10}(100) + \log_{10}(1000)

• Both have the same base of 10.
• Multiply the values inside: 100 \times 1000 = 100000.
• So, \log_{10}(100) + \log_{10}(1000) = \log_{10}(100000).
• \log_{10}(100000)=5 because 10^5=100000.
• Answer: 5.

Adding more than two logarithms

We can do exactly the same when we are adding more than two logarithms with the same base - just multiply all the values inside.

Example: Evaluate \log_3(9) + \log_3(27) + \log_3(81)

• All have the same base of 3.
• Multiply the values inside: 9 \times 27 \times 81 = 19683.
• So, \log_3(9) + \log_3(27) + \log_3(81) = \log_3(19683).
• \log_3(19683)=9 because 3^9=19683.
• Answer: 9.

Splitting up a logarithm

You can also split up a logarithm into the sum of two logarithms:

\log_b(xy) = \log_b(x) + \log_b(y)

Example: write \log_5(1250) as the sum of an integer and a logarithm

• 1250 can be split into 125 \times 10.
• So, \log_5(1250) = \log_5(125) + \log_5(10).
• \log_5(125)=3 because 5^3=125.
• \log_5(10) cannot be simplified further.
• So, \log_5(1250) = 3 + \log_5(10).
• Answer: 3 + \log_5(10).