Approximate Logarithms with Technology Solutions

• Use technology to    approximate    the value of logs.

• If needed, use    logarithmic properties    to expand log expressions in order to approximate their value. 

• To avoid errors, only round as the    very last    step.

Example 3

Write $\mathbf{log}\mathbf{ }\mathbf{24}$ in terms of X and Y when $\mathbf{log}\mathbf{ }\mathbf{2}\mathbf{=}\mathit{X}$ and $\mathbf{log}\mathbf{ }\mathbf{3}\mathbf{=}\mathit{Y}$.

Example 4

1. Write $\mathbf{log}\mathbf{ }\mathbf{36}$ in terms of X and Y when $\mathbf{log}\mathbf{ }\mathbf{2}\mathbf{=}\mathit{X}$ and $\mathbf{log}\mathbf{ }\mathbf{3}\mathbf{=}\mathit{Y}$.
2. Calculate the approximate value of $\mathbf{log}\mathbf{ }\mathbf{36}$ when $\mathbf{log}\mathbf{ }\mathbf{2}\mathbf{\approx }\mathbf{0}\mathbf{.}\mathbf{301}$ and $\mathbf{log}\mathbf{ }\mathbf{3}\mathbf{\approx }\mathbf{0}\mathbf{.}\mathbf{4771}$. 

Plan

Express the argument as its prime factorization


Expand with the rules for logs


Substitute in the given values

Check

Note

Problems A–B

Use technology to find the approximate value of $\log 36$ and the expanded log.

 

Lesson 39 More to Explore shows how to use technology to work with approximating logs.

1. $\begin{array}{rcl}\log 36& =& \log \left(2^2·3^2\right)\\ & =& \log 2^2+\log 3^2\\ & =& 2\log 2+2 \log 3\\ & =& 2\left(X\right)+2\left(Y\right)\\ & =& 2X+2Y\end{array}$ 

2. $\begin{array}{rcl}\log 36& =& 2X+2Y\\ & \approx & 2\left(0.301\right)+2\left(0.4771\right)\\ & \approx & 1.5562\end{array}$