Consider the following exponential function of value of the home:

\(\displaystyle{V}{\left({t}\right)}={a}{b}^{{t}}\)

Where,

V(t)=Value of home after t years

t=number of years from 2012

a and b are constants

In 2012, value of home is $236,500

\(\displaystyle{a}{b}^{{0}}={236500}\)

\(\displaystyle{a}={236500}\)

In 2017, value of home is $305,700

\(\displaystyle{t}={2017}-{2012}\)

\(\displaystyle={5}\)

\(\displaystyle{236500}\times{b}^{{5}}={305700}\)

\(\displaystyle{b}^{{5}}={\frac{{{305700}}}{{{236500}}}}\)

\(\displaystyle{b}^{{5}}={1.2926}\)

\(\displaystyle{b}=\sqrt{{{5}}}{\left\lbrace{1.2926}\right\rbrace}\)

\(\displaystyle{b}={1.05267}\)

Hence, the exponential function is \(\displaystyle{V}{\left({t}\right)}={236500}\times{\left({1.05267}\right)}^{{t}}={400000}\)

\(\displaystyle{\left({1.05267}\right)}^{{t}}={\frac{{{400000}}}{{{236500}}}}\)

\(\displaystyle{\left({1.05267}\right)}^{{t}}={1.691332}\)

Take logarithm on both sides of the equation:

\(\displaystyle{t}\times{\log{{\left({1.05267}\right)}}}={\log{{\left({1.691332}\right)}}}\)

\(\displaystyle{t}={\frac{{{\log{{\left({1.691332}\right)}}}}}{{{\log{{\left({1.05267}\right)}}}}}}\)

\(\displaystyle={\frac{{{0.228229}}}{{{0.02229}}}}\)

\(\displaystyle={10.23}\)

\(\displaystyle\approx{10}\)

Hence, after 10 years value of home will become $400,000.

\(\displaystyle{V}{\left({t}\right)}={a}{b}^{{t}}\)

Where,

V(t)=Value of home after t years

t=number of years from 2012

a and b are constants

In 2012, value of home is $236,500

\(\displaystyle{a}{b}^{{0}}={236500}\)

\(\displaystyle{a}={236500}\)

In 2017, value of home is $305,700

\(\displaystyle{t}={2017}-{2012}\)

\(\displaystyle={5}\)

\(\displaystyle{236500}\times{b}^{{5}}={305700}\)

\(\displaystyle{b}^{{5}}={\frac{{{305700}}}{{{236500}}}}\)

\(\displaystyle{b}^{{5}}={1.2926}\)

\(\displaystyle{b}=\sqrt{{{5}}}{\left\lbrace{1.2926}\right\rbrace}\)

\(\displaystyle{b}={1.05267}\)

Hence, the exponential function is \(\displaystyle{V}{\left({t}\right)}={236500}\times{\left({1.05267}\right)}^{{t}}={400000}\)

\(\displaystyle{\left({1.05267}\right)}^{{t}}={\frac{{{400000}}}{{{236500}}}}\)

\(\displaystyle{\left({1.05267}\right)}^{{t}}={1.691332}\)

Take logarithm on both sides of the equation:

\(\displaystyle{t}\times{\log{{\left({1.05267}\right)}}}={\log{{\left({1.691332}\right)}}}\)

\(\displaystyle{t}={\frac{{{\log{{\left({1.691332}\right)}}}}}{{{\log{{\left({1.05267}\right)}}}}}}\)

\(\displaystyle={\frac{{{0.228229}}}{{{0.02229}}}}\)

\(\displaystyle={10.23}\)

\(\displaystyle\approx{10}\)

Hence, after 10 years value of home will become $400,000.