If log _{10} 2, log _{10} (2^x + 1), log _{10 | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

If $a, b, c$ are in $AP$ then $2 b=a+c$

$\Rightarrow 2 \log _{10}\left(2^{x}-1\right)=\log _{10} 2+\log _{10}\left(2^{x}+3\right)$

$\Rightarrow

Vedclass · Accepted Answer

$x = \frac{1}{2} \log _2 5$

Vedclass · Answer

If $a, b, c$ are in $AP$ then $2 b=a+c$

$\Rightarrow 2 \log _{10}\left(2^{x}-1\right)=\log _{10} 2+\log _{10}\left(2^{x}+3\right)$

$\Rightarrow \log _{10}\left(2^{x}-1\right)^{2}=\log _{10} 2\left(2^{x}+3\right)$

$\Rightarrow \log _{10}\left(2^{2 x}+1-2^{x+1}\right)=\log _{10}\left(2^{x+1}+6\right)$

$\Rightarrow 2^{2 x}+1-2^{x+1}=2^{x+1}+6$

$\Rightarrow 2^{2 x}-2^{x+2}-5=0$

Take, $2^{x}=y$

$\Rightarrow y^{2}-4 y-5=0$

$\Rightarrow(y-5)(y+1)=0$

$\Rightarrow y=5,-1$

$\because y>0 \Rightarrow y=5$

$\Rightarrow 2^{x}=5$

$\Rightarrow x=\log _{2} 5$

If $\log _{10} 2, \log _{10} (2^x + 1), \log _{10} (2^x + 3)$ are in $A.P.,$ then :-

Similar Questions

The solution of the equation $(x + 1) + (x + 4) + (x + 7) + ......... + (x + 28) = 155$ is

Let ${\left( {1 - 2x + 3{x^2}} \right)^{10x}} = {a_0} + {a_1}x + {a_2}{x^2} + .....+{a_n}{x^n},{a_n} \ne 0$, then the arithmetic mean of $a_0,a_1,a_2,...a_n$ is

In an $\mathrm{A.P.}$ if $m^{\text {th }}$ term is $n$ and the $n^{\text {th }}$ term is $m,$ where $m \neq n$, find the ${p^{th}}$ term.

If the first term of an $A.P. $ be $10$, last term is $50$ and the sum of all the terms is $300$, then the number of terms are

The income of a person is $Rs. \,3,00,000,$ in the first year and he receives an increase of $Rs.\,10,000$ to his income per year for the next $19$ years. Find the total amount, he received in $20$ years.