What is the value of x in the following expression?
$$x + \log_{10} (1 + 2^x) = x \log_{10} 5 + \log_{10} 6$$
The given equation can be written asÂ
$$\log\left(10\right)^{x\ }\ +\ \log\left(1+2^x\right)=\log\left(5\right)^x+\log6$$
$$\log\left(10\right)^{x\ }\left(1+2^x\right)=\log\left(5\right)^x\cdot6$$Â Â (Â since logA + logB=logAB)
$$\log\ \frac{\left(2^x\cdot5^x\right)\left(1+2^x\right)}{5^x\cdot6}=0$$Â Â Â ( since logA - logB=logA/B)
$$\frac{\left(2^x\ +2^{2x\ }\right)}{6}=10^0$$Â ($$Since\ \log_aN\ =x\ \ =>N=a^x$$)
$$2^{^x}+2^{2x}=6$$
The above  equation is satisfied only when x=1
Create a FREE account and get: