Convert to the logarithmic equation that has the same meaning.
[tex]g^d=34[/tex]
The equation is $\square$ .
(Use integers or fractions for any numbers in the equation.)
Answer (2)
The problem requires converting an exponential equation to its logarithmic form.
Recall the general conversion: b x = y is equivalent to lo g b y = x .
Apply this to the given equation g d = 34 , identifying b = g , x = d , and y = 34 .
The logarithmic form of the equation is lo g g 34 = d .
Explanation
Understanding the Problem We are given the exponential equation g d = 34 and asked to convert it to its equivalent logarithmic form.
Recalling the Conversion Recall that the exponential equation b x = y is equivalent to the logarithmic equation lo g b y = x . In our case, we have g d = 34 , so we can identify b = g , x = d , and y = 34 .
Applying the Conversion Applying the conversion, we get lo g g 34 = d .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes using the Richter scale, determining the pH levels in chemistry, and modeling population growth in biology. For instance, if we know the growth rate of a bacteria population and want to find out how long it takes for the population to reach a certain size, we can use logarithmic equations to solve for the time. Similarly, in finance, logarithmic equations are used to calculate the time it takes for an investment to double at a given interest rate.
The logarithmic form of the equation g d = 34 is lo g g 34 = d . This conversion follows the standard form where if b x = y , then lo g b y = x . Identifying base, exponent, and result allows for a seamless transition between forms.
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