91Ó°ÊÓ

Let \(b\) be any positive real number such that \(b \neq 1\). What must \(\log _{b} 1\) be equal to? Verify the result.

Explore and discuss the graphs of \(f(x)=\log _{1}(x)\) and \(g(x)=-\log _{2}(x) .\) Make a conjecture based on the result.

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. $$\log (\sqrt{2})$$

For the following exercises, use a graphing calculator to find approximate solutions to each equation.$$\frac{1}{3} \log (1-x)=\log (x+1)+\frac{1}{3}$$

Use a graphing utility to fi \(\mathrm{d}\) an exponential regression formula \(f(x)\) and a logarithmic regression formula \(\mathrm{g}(x)\) for the points (1.5,1.5) and (8.5,8.5) Round all numbers to 6 decimal places. Graph the points and both formulas along with the line \(y=x\) on the same axis. Make a conjecture about the relationship of the regression formulas.

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. $$ \ln (4 x-10)-6=-5 $$

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. $$\ln (\sqrt{2})$$

Let \(b\) be any positive real number such that \(b \neq 1\) . What must log, 1 be equal to? Verify the result.

Recall that an exponential function is any equation written in the form \(f(x)=a \cdot b^{x}\) such that \(a\) and \(b\) are positive numbers and \(b \neq 1\) . Any positive number \(b\) can be written as \(b=e^{n}\) for some value of \(n .\) Use this fact to rewrite the formula for an exponential function that uses the number \(e\) as a base.

What is the domain of the function \(f(x)=\ln \left(\frac{x+2}{x-4}\right) ?\) Discuss the result.