How do you find the exponential function f(x)= a^x whose graph goes through the point (3, 1/125)?

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Answer 1 · 2016-12-06T18:12:53

Please see the explanation.

Use the logarithm of your favorite base on the function (My favorite is the natural logarithm but base 10 works just as well):

#ln(f(x)) = ln(a^x)#

Use the identity ln(a^x) = (x)ln(a):

#ln(f(x)) = (x)ln(a)#

Divide both sides by x and flip the equation:

#ln(a) = ln(f(x))/x#

Make both sides exponents of the base e:

#e^ln(a) = e^(ln(f(x))/x)#

Use the identity #e^ln(a) = a#

#a = e^(ln(f(x))/x)#

Substitute 3 for x and 1/125 for #f(x)#:

#a = e^(ln(1/125)/3)#

#a = 0.2#

If you prefer a fraction:

#a = 1/5#