To convert the exponential equation 3x = 243 into logarithmic form, we need to understand how to express exponential equations in terms of logarithms. The general form for this conversion is:
bx = a can be written as logb a = x, where b is the base of the exponent, a is the result, and x is the exponent.
In our specific equation, we identify:
- b = 3 (the base),
- a = 243 (the result),
- x (the exponent that we want to express as a logarithm).
Thus, applying the formula, we have:
log3 243 = x
However, since the question specifies to express this in base 10, we can use the change of base formula, which allows us to convert logarithms from one base to another:
logb a = logk a / logk b
Applying this to our equation:
log3 243 = log10 243 / log10 3
This means the logarithmic form of the equation 3x = 243 in base 10 can be expressed as:
x = log10 243 / log10 3
In conclusion, the logarithmic form of the given exponential equation in base 10 is:
x = log10 243 / log10 3
This transformation is essential in solving for values of x in various applications, especially in fields such as science and engineering, where such equations frequently arise.