solve the following equation. \n$5^{5x - 4} = 10$\n$x = \\square$ (type an exact answer.)

Explanation:

Step1: Take log on both sides

Take the logarithm (natural log or common log, here we use natural log) of both sides of the equation (5^{5x - 4}=10). We get (\ln(5^{5x - 4})=\ln(10)). Using the logarithm power rule (\ln(a^b)=b\ln(a)), the left - hand side becomes ((5x - 4)\ln(5)=\ln(10)).

Step2: Solve for x

First, divide both sides of the equation ((5x - 4)\ln(5)=\ln(10)) by (\ln(5)): (5x-4 = \frac{\ln(10)}{\ln(5)}) Then, add 4 to both sides: (5x=\frac{\ln(10)}{\ln(5)} + 4) Finally, divide both sides by 5: (x=\frac{\frac{\ln(10)}{\ln(5)}+4}{5}=\frac{\ln(10)+4\ln(5)}{5\ln(5)}) We can also simplify (\frac{\ln(10)+4\ln(5)}{5\ln(5)}) using the property (a\ln(b)=\ln(b^a)) and (\ln(a)+\ln(b)=\ln(ab)). Since (4\ln(5)=\ln(5^4)=\ln(625)) and (\ln(10)+\ln(625)=\ln(10\times625)=\ln(6250)), so (x = \frac{\ln(6250)}{5\ln(5)}) or (x=\frac{\log_5(10)+4}{5}) (using the change - of - base formula (\frac{\ln(a)}{\ln(b)}=\log_b(a))).

Answer:

(\frac{\log_{5}(10)+4}{5}) (or equivalent forms like (\frac{\ln(10)+4\ln(5)}{5\ln(5)}) or (\frac{\ln(6250)}{5\ln(5)}))