Select the correct answer. Which exponential equation is equivalent to this logarithmic equation? \( \log _{x} 5+\log _{x} 12=7 \) A. \( x^{7}=17 \) B. \( 7^{x}=60 \) C. \( 7^{x}=17 \) D. \( x^{7}=60 \)

To convert the logarithmic equation \( \log _{x} 5+\log _{x} 12=7 \) into an exponential form, we can combine the logarithms on the left side using the property \( \log_a b + \log_a c = \log_a (bc) \). This gives us \( \log_x (5 \times 12) = 7\), or \( \log_x 60 = 7 \). Converting this logarithmic form to exponential gives us \( x^7 = 60 \). Thus, the correct answer is D. Real-World Application: Understanding logarithms and exponents is crucial in fields like finance, where exponential growth is commonly observed with investments. When calculating how long it will take for money to double with a given interest rate, you often use logarithmic equations to solve for time, making comprehension of these concepts key to financial literacy! Problem-Solving Tips & Common Mistakes: A common mistake when working with logarithmic equations is forgetting to apply the properties correctly or misapplying the base of the logarithm. Always double-check which base you are working with and remember to convert the expression fully into its exponential form before solving for the variable, as it helps prevent confusion down the road!