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Chapter 4: Problem 34
Find the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) for each substance with the given \(p H\) wine, 3.4
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Let \(u=\ln a\) and \(v=\ln b .\) Write each expression in terms of \(u\) and \(v\) without using the In function. $$\ln \left(\sqrt[3]{a} \cdot b^{4}\right)$$
Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts \((a)-(c)\) $$\begin{aligned} &\text { Given } f(x)=\log _{2} x, \text { find } \quad \text { (a) } f\left(2^{7}\right) \quad \text { (b) } f\left(2^{\log _{2} 2}\right)\\\ &\text { (c) } f\left(2^{2 \log _{3} 2}\right) \end{aligned}$$
In calculus, it is shown that $$ e^{x}=1+x+\frac{x^{2}}{2 \cdot 1}+\frac{x^{3}}{3 \cdot 2 \cdot 1}+\frac{x^{4}}{4 \cdot 3 \cdot 2 \cdot 1}+\frac{x^{5}}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}+\cdots $$ By using more terms, one can obtain a more accurate approximation for \(e^{x}\). Use the terms shown, and replace \(x\) with \(-0.05\) to approximate \(e^{-0.05}\) to four decimal places. Check your result with a calculator.
Use a graphing calculator to find the solution set of each equation. Give solutions to the nearest hundredth. $$2^{-x}=\log _{10} x$$
For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=\frac{4}{x}, \quad x \neq 0$$
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