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Chapter 3: Problem 19

Given \(\log _{b} 3=1.099\) and \(\log _{b} 5=1.609,\) find each value. $$\log _{b} 15$$

Short Answer

Expert verified
The value is \log_{b}(15) = 2.708.

Step by step solution

01

Apply the Product Rule of Logarithms

Use the product rule for logarithms, which states: \[\forall a, b, c > 0, \, \log_{b}(ac) = \log_{b}(a) + \log_{b}(c) \]In this case, we need to find \log_{b}(15). Since 15 can be expressed as a product of 3 and 5, where 15 = 3 \cdot 5, we can write:\[\log_{b}(15) = \log_{b}(3 \cdot 5) = \log_{b}(3) + \log_{b}(5) \]
02

Substitute Given Values

Use the provided values of \log_{b}(3) and \log_{b}(5) to substitute into the formula obtained in Step 1:\[\log_{b}(3) = 1.099 \]\[\log_{b}(5) = 1.609 \]Substituting these values into the equation:\[\log_{b}(15) = 1.099 + 1.609 \]
03

Simplify the Expression

Add the values obtained in Step 2 to get the final result:\[\log_{b}(15) = 1.099 + 1.609 = 2.708 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule
The product rule for logarithms is essential for breaking down more complex logarithmic expressions into simpler parts. It states that for any positive values of a, b, and c, \[ \log_{b}(a \cdot c) = \log_{b}(a) + \log_{b}(c) \]. This means that the logarithm of a product is the sum of the logarithms of the factors. For example, in the given exercise, we have \[ \log_{b}(15) \]. Since 15 can be factored into 3 and 5, we use the product rule: \[ \log_{b}(15) = \log_{b}(3 \cdot 5) = \log_{b}(3) + \log_{b}(5) \]. This simplifies the original problem to two separate, simpler logarithmic expressions that we can handle more easily.
Logarithmic Identities
Logarithmic identities are crucial for transforming and simplifying logarithmic expressions. One of the most frequently used identities is the product rule, but there are others like the quotient rule and the power rule. The product rule, as shown in our exercise, states that \[ \log_{b}(a \cdot c) = \log_{b}(a) + \log_{b}(c) \].

**Other Useful Logarithm Identities**
  • **Quotient Rule:** \[ \log_{b}(\frac{a}{c}) = \log_{b}(a) - \log_{b}(c) \] This rule helps when dealing with a division inside a logarithm.
  • **Power Rule:** \[ \log_{b}(a^c) = c \cdot \log_{b}(a) \] This rule simplifies logarithms of numbers raised to a power.
Understanding these identities can immensely simplify seemingly complex logarithmic problems.
Logarithm Simplification
Logarithm simplification involves using various rules and identities to break down or combine logarithmic expressions. By applying these rules, we can move from a complex expression toward a simpler or more solvable form.

In our exercise, we simplified \[ \log_{b}(15) \] by expressing 15 as a product of 3 and 5: \[ \log_{b}(3 \cdot 5) = \log_{b}(3) + \log_{b}(5) \]. Next, we substituted the given values: \[ \log_{b}(3) = 1.099 \] and \[ \log_{b}(5) = 1.609 \], which gave us: \[ \log_{b}(15) = 1.099 + 1.609 = 2.708 \].

Properly using these simplification techniques can make solving logarithmic expressions much more manageable.

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