Logarithm Functions — Question 8

Question 8

Let $f(x)=\log_b x$, where $b>0$ and $b\ne1$. Suppose that $$f(2)=1\qquad\text{and}\qquad f(5)\approx2.32193.$$

• Estimate $f(10)$ using logarithm properties.

• Estimate $f(5/2)$ using logarithm properties.

• Explain the properties used.

Question 8 - Solution

The product and quotient laws hold for positive arguments:

$$f(10)=f(2)+f(5)\approx1+2.32193=\boxed{3.32193},$$

$$f(5/2)=f(5)-f(2)\approx2.32193-1=\boxed{1.32193}.$$

The datum $\log_b2=1$ fixes $b=2$, so the supplied approximation to $\log_b5$ is consistent with the same base.