Rewrite the following exponential functions as logarithmic functions, and the logarithmic functions as exponential functions
a)
f(x)=(7/8)^x
b)
f(x)=log12x
c)
f(x)=log5/9x
y = (7/8)^x
log(y) = x log(7/8)
...
y=log12x
10^y = 12x
...
and similarly for the last one.
a) Expotential Function to Logarithmic Function:
f(x) = (7/8)^x
To rewrite this exponential function in logarithmic form, we use the following relationship:
y = a^x => x = log_a(y)
Applying this to the given function, we have:
x = log_(7/8)(f(x))
Therefore, the logarithmic function is:
f(x) = log_(7/8)(f(x))
b) Logarithmic Function to Exponential Function:
f(x) = log12(x)
To rewrite this logarithmic function in exponential form, we use the following relationship:
y = log_a(x) => x = a^y
Applying this to the given function, we have:
x = 12^f(x)
Therefore, the exponential function is:
f(x) = 12^f(x)
c) Logarithmic Function to Exponential Function:
f(x) = log(5/9)x
To rewrite this logarithmic function in exponential form, we use the following relationship:
y = log_a(x) => x = a^y
Applying this to the given function, we have:
x = (5/9)^f(x)
Therefore, the exponential function is:
f(x) = (5/9)^f(x)