Logarithms

Logarithms Intro

log_b(a) = c

b^c = a

log₂(8) = 3

2³ = 8

How many b’s need to be multiplied together to get a?

Natural Logarithms and e

e is an irrational number equal to 2.71828… and is commonly known as Euler’s number or Napier’s constant. It is the base of what is known as the natural logarithm. The natural logarithm is like the usual common logarithm except for one subtle difference: it has the irrational constant e as its base instead of 10. To make it easier for mathematicians to write down, the natural logarithm is denoted as “ln” while the common logarithm is denoted as “log”.

Natural Log (base e) and Common Log (base 10)

When you are told to evaluate ln x, it is implied that the base is e. For example, if you are told to evaluate ln e², your thought process should go like this:

e to the what is e²? The answer should be 2.

Natural log problems usually require the use of a calculator.

Why?

e is an irrational number (2.718281828459045…)

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Natural Log (base e) and Common Log (base 10)

When you are told to evaluate log x, it is implied that the base is 10. For example, if you are told to evaluate log 100, your thought process should go like this:

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Check your answer by converting to the exponential form of a logarithm:

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Since this is true, the answer 2 is correct.

log₁₀(10²) = 2log₁₀(10) = 2(1) = 2

log₁₀(100) = 2 ⟶ 10² = 100

y = log x

y = log x

y = 10^x

y = ln(x)

y = log_ex

y = ln(x)

y = e^x

y = ln(x)

y = log x

y = e^x

y = 10^x

Questions

1. Given the log of 5=0.699, what is the log of 500?

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2. Find y if log₈y = -3

3. If ln 4 = 1.386 and ln 5 = 1.609, what is ln 1.25?

Question #1

1. If log₁₀5 = 0.699, then what is log₁₀500?

log₁₀500 = log₁₀5*10² = log₁₀5 + log₁₀100 = 0.699+2= 2.699

Question #2

1. Find y if log₈y = -3

Convert to the exponential form to get the question into _____ = y form.

8^-3 = y

⅛³ = y

1/512 = y

Question #3

1. If ln 4 = 1.386 and ln 5 = 1.609, what is ln 1.25?

Use the rule log x - log y = log (x/y)

5/4= 1.25, so ln (5/4) = ln 1.25

Since ln (5/4) is in the form log (x/y), it can be rewritten as ln 5 - ln 4.

Now plug in the values for ln 5 - ln 4.

1.609 - 1.386 = 0.223