Which exponential equation is equivalent to this logarithmic equation?
$\log _7 6+3 \log _7 2=x$
The objective is to find an exponential equation equivalent to the logarithmic equation \(\log_{7} 6 + 3\log_{7} 2 = x\). To solve this, we rely on the fundamental properties of logarithms, specifically the power rule and the product rule, and the definition of a logarithm that relates logarithmic forms to exponential forms.
The power rule states that \(n \log_b M = \log_b (M^n)\). We can apply this to the term \(3 \log_{7} 2\):
\[3 \log_{7} 2 = \log_{7} (2^3) = \log_{7} 8\]
By simplifying the expression inside the logarithm, we get \(\log_{7} 8\).
Next, we use the product rule, which states that \(\log_b M + \log_b N = \log_b (MN)\). Applying this to the entire left side of the original equation:
\[\log_{7} 6 + \log_{7} 8 = \log_{7} (6 \times 8)\]
Calculating the product \(6 \times 8 = 48\), the equation simplifies to:
\[\log_{7} 48 = x\]
The definition of a logarithm states that \(\log_b Y = x\) is equivalent to \(b^x = Y\). Using this definition, we can convert our simplified logarithmic equation \(\log_{7} 48 = x\) into its equivalent exponential form:
\[7^x = 48\]
Comparing this to the given options, we find that \(7^x = 48\) is the correct equivalent equation.
Logarithms are essential in measuring phenomena that vary across huge ranges, such as the Richter scale for earthquake magnitude or the pH scale in chemistry. For example, if you are determining the growth of an investment using continuously compounded interest, you might start with a logarithmic relationship to solve for time or growth rates. Converting between logarithmic and exponential forms allows scientists and economists to isolate the variables they need to solve for, making complex calculations manageable.