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Use exponential rules to write an equivalent exponential function with only \( x \) in the exponent. Answers should be of the form \( G(x)=a\left(b^{x}\right) \). Express your answers using fractions. \( G(x)=4^{6 x-1} \)

Convert the temperature, as indicated. Round your answer to hundredths, if appropriate. 7) 370 K , into Celsius 8) \( 45^{\circ} \mathrm{C} \), into Fahrenheit

Solve the following equation. Express your answer as either an integer or a fraction in lowest terms. \( 81^{-4 x+5}=\left(\frac{1}{3}\right)^{3 x-2} \)

29. Let \[ A=\left[\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 1\end{array}\right] \] We wish to determine all \( 4 \times 3 \) matrices \( X \) for which \( A X A=A \) Observe that \[ P A Q=\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right] \] where \( P \) and \( Q \) are the invertible matrices \[ P=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 2 & 1\end{array}\right] \text { and } Q=\left[\begin{array}{rrrr}1 & -1 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1\end{array}\right] . \] (a) Explain how you can use this to convert the problem of finding all \( X \) so that \( A X A=A \) to finding all \( 4 \times 3 \) matrices \( Y \) so that \( B Y B=B \), where \( B=P A Q \). Make sure to indicate how \( X \) and \( Y \) are related. (b) Find all \( Y \) so that \( B Y B=B \). (c) Use your answer in (b) to find all \( X \) so that \( A X A=A \). (d) Give 2 explicit \( X \) with \( A X A=A \).

Use the definition of the logarithms to evaluate the following logarithms. a. \( \log \left(\frac{1}{10}\right)=\square \) b. \( \log (10000000)=\square \) c. \( \log (100000)=\square \)