log _(a) t=(lg n)/(lg theta_(1) s)+log _(4) 299

To solve the equation \( \log _{a} t=\frac{\lg n}{\lg \theta_{1} s}+\log _{4} 299 \), we need to simplify the expression on the right-hand side.<br /><br />First, let's rewrite the logarithmic terms using the change of base formula:<br /><br />\[<br />\log_{a} t = \frac{\log_{10} n}{\log_{10} (\theta_{1} s)} + \log_{4} 299<br />\]<br /><br />Next, we can express \(\log_{4} 299\) in terms of common logarithms:<br /><br />\[<br />\log_{4} 299 = \frac{\log_{10} 299}{\log_{10} 4}<br />\]<br /><br />Now, substitute this back into the original equation:<br /><br />\[<br />\log_{a} t = \frac{\log_{10} n}{\log_{10} (\theta_{1} s)} + \frac{\log_{10} 299}{\log_{10} 4}<br />\]<br /><br />Combine the fractions on the right-hand side:<br /><br />\[<br />\log_{a} t = \frac{\log_{10} n + \frac{\log_{10} 299}{\log_{10} 4}}{\log_{10} (\theta_{1} s)}<br />\]<br /><br />Simplify the numerator:<br /><br />\[<br />\log_{a} t = \frac{\log_{10} n + \log_{10} 299}{\log_{10} (\theta_{1} s) \cdot \log_{10} 4}<br />\]<br /><br />Combine the logarithms in the numerator:<br /><br />\[<br />\log_{a} t = \frac{\log_{10} (n \cdot 299)}{\log_{10} (\theta_{1} s) \cdot \log_{10} 4}<br />\]<br /><br />Finally, we can rewrite the equation in exponential form:<br /><br />\[<br />t = a^{\frac{\log_{10} (n \cdot 299)}{\log_{10} (\theta_{1} s) \cdot \log_{10} 4}}<br />\]<br /><br />This is the simplified form of the given equation.