How do you solve 4^x * 5^(4x+3) = 10^(2x+3)?

2 Answers
Mar 28, 2016

x~~0.65

Explanation:

1. Start by taking the logarithm of both sides, since the bases are not the same on the left and right sides of the equation.

4^x*5^(4x+3)=10^(2x+3)

log(4^x*5^(4x+3))=log(10^(2x+3))

2. Use the log property, log_color(purple)b(color(red)m*color(blue)n)=log_color(purple)b(color(red)m)+log_color(purple)b(color(blue)n) to simplify the left side of the equation.

log(4^x)+log(5^(4x+3))=log(10^(2x+3))

3. Use the log property, log_color(purple)b(color(red)m^color(blue)n)=color(blue)n*log_color(purple)b(color(red)m), to rewrite both sides of the equation.

xlog(4)+(4x+3)log(5)=(2x+3)log(10)

4. Expand the brackets.

xlog(4)+4xlog(5)+3log(5)=2xlog(10)+3log(10)

5. Bring all terms with the variable, x, to the left side of the equation and all terms without to the right side.

xlog(4)+4xlog(5)-2xlog(10)=3log(10)-3log(5)

6. Factor out x from the terms on the left side.

x(log(4)+4log(5)-2log(10))=3log(10)-3log(5)

7. Solve for x.

x=(3log(10)-3log(5))/(log(4)+4log(5)-2log(10))

color(green)(|bar(ul(color(white)(a/a)x~~0.65color(white)(a/a)|)))

Apr 16, 2016

0.646

Explanation:

4^x*5^(4x+3)=10^(2x+3)

=>4^x*5^(4x)*5^3=10^(2x)*10^3

=>(4*5^4)^x*5^3=(10^2)^x*10^3

Dividing both sides by 5^3*(10^2)^x

=>((4*5^4)/10^2)^x=10^3/5^3

=>((cancel4*cancel5xxcancel5xx5^2)/(cancel10xxcancel10))^x=1000/125=8=2^3

Taking log_10 on both sides and using property logm^n=nlogm we get
=>log_10 5^(2x)=log_10 2^3
=>2x*log_10 5=3*log_10 2

Dividing both sides by (2*log_10 5)

=>x=(3*log_10 2)/(2*log_10 5)=0.646