Solve (a) \(10^{x}=7\) (b) \(10^{x}=70\) (c) \(10^{x}=17\) (d) \(10^{\mathrm{x}}=0.7000\)

To rewrite an exponential equation with base 10 as a logarithmic equation, we can use the formula: \(\log_{a}^{b} = x\), where a is the base, b is the exponent, and x is the result. (a) \(10^{x}=7\) \(\log_{10}(7)=x\) (b) \(10^{x}=70\) \(\log_{10}(70)=x\) (c) \(10^{x}=17\) \(\log_{10}(17) = x\) (d) \(10^{\mathrm{x}}=0.7000\) \(\log_{10}(0.7000) = x\)

To solve for x, we will use log properties, specifically the change of base formula to calculate the base 10 logarithms. Change of base formula: \(\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}\) For the base 10 logarithm, we will change the base to the natural logarithm (base e) because it is easily available on most calculators. The change of base formula will be as follows: \(\log_{10}(b) = \frac{\ln(b)}{\ln(10)}\) (a) \(x = \log_{10}(7) = \frac{\ln(7)}{\ln(10)} \approx 0.8451\) (b) \(x = \log_{10}(70) = \frac{\ln(70)}{\ln(10)} \approx 1.8451\) (c) \(x = \log_{10}(17) = \frac{\ln(17)}{\ln(10)} \approx 1.2304\) (d) \(x = \log_{10}(0.7000) = \frac{\ln(0.7000)}{\ln(10)}\approx -0.1549\) The solutions for each part are: (a) \(x\approx 0.8451\), (b) \(x\approx 1.8451\), (c) \(x\approx 1.2304\), and (d) \(x\approx -0.1549\).