Questions: Write the equation in logarithmic form. Assume that all constants are positive and not equal to 1. 9^u=a

Transcript text: Write the equation in logarithmic form. Assume that all constants are positive and not equal to 1. \[ 9^{u}=a \]

Solution

Solution Steps

Step 1: Given Exponential Equation

We start with the exponential equation: \[ 9^{u} = a \]

Step 2: Convert to Logarithmic Form

To convert the exponential equation to logarithmic form, we use the base of the exponent (which is 9) as the base of the logarithm, the exponent \( u \) as the value of the logarithm, and the result of the exponentiation \( a \) as the argument of the logarithm. This gives us: \[ u = \log_{9}(a) \]

Step 3: Express in Terms of Natural Logarithm

Using the change of base formula, we can express the logarithm in terms of natural logarithms: \[ u = \frac{\log(a)}{\log(9)} \]