If \( \log _{2}(5 x+6)=5 \), then \( x=\square \) You may enter the exact value or round to 4 decimal places. Question Help: Submit Question

To solve the equation \( \log_{2}(5x + 6) = 5 \), begin by rewriting it in exponential form, which gives us \( 5x + 6 = 2^5 \). Since \( 2^5 = 32 \), this simplifies to \( 5x + 6 = 32 \). Next, subtract 6 from both sides: \( 5x = 32 - 6 \), resulting in \( 5x = 26 \). Finally, divide both sides by 5 to isolate \( x \): \[ x = \frac{26}{5} = 5.2 \] So, the value of \( x \) is \( 5.2 \). Delving into the roots of logarithmic equations reveals that they were essential for early mathematicians dealing with exponential growth. In fact, logarithms were invented in the 16th century as a tool for simplifying complex calculations, especially for astronomers and navigators who relied heavily on precise measurements. For practical applications, understanding logarithms can be invaluable, especially in fields like finance, where interest calculations often involve exponential growth. If you’re ever grappling with equations involving logs, remember that converting to exponential form can simplify the problem significantly, allowing you to tackle complex applications with ease!