Question

How do you solve \[\log \left( {2x + 1} \right) - \log \left( {x - 2} \right) = 1\]?

Answer 1

Hint: To find the value of \[\log \left( {2x + 1} \right) - \log \left( {x - 2} \right) = 1\], we will use the quotient rule i.e., \[{\log _b}\dfrac{m}{n} = {\log _b}m - {\log _b}n\] to simplify it. Then using the property of \[{\log _b}\left( {{b^a}} \right) = a\] we will rewrite \[1\] as \[\log \left( {10} \right)\]. If bases are the same on both sides of an equation then the equation can be written without \[\log \]. Using this we will write an equation in \[x\] and then we will simplify it.

Complete step by step answer:
We have to solve \[\log \left( {2x + 1} \right) - \log \left( {x - 2} \right) = 1\].
Now using the quotient rule i.e., \[{\log _b}\dfrac{m}{n} = {\log _b}m - {\log _b}n\] we can write
\[ \Rightarrow \log \left( {\dfrac{{2x + 1}}{{x - 2}}} \right) = 1\]
Now, as we know from the property of \[\log \] that \[{\log _b}\left( {{b^a}} \right) = a\]. Using this, we get
\[ \Rightarrow \log \left( {\dfrac{{2x + 1}}{{x - 2}}} \right) = \log \left( {10} \right)\]
As both sides of the equation contain \[\log \] and bases are also the same on both sides of the equation. So, the equation can also be written as
\[ \Rightarrow \dfrac{{2x + 1}}{{x - 2}} = 10\]
On cross multiplication, we get
\[ \Rightarrow 2x + 1 = 10\left( {x - 2} \right)\]
On simplification, we get
\[ \Rightarrow 2x + 1 = 10x - 20\]
Subtracting \[2x\] from both the sides, we get
\[ \Rightarrow 1 = 10x - 2x - 20\]
Adding \[20\] on both the sides, we get
\[ \Rightarrow 1 + 20 = 10x - 2x\]
On simplification and rearranging, we get
\[ \Rightarrow 8x = 21\]
Dividing both the sides by \[8\], we get
\[ \Rightarrow x = \dfrac{{21}}{8}\]
Therefore, on solving \[\log \left( {2x + 1} \right) - \log \left( {x - 2} \right) = 1\], we get \[x = \dfrac{{21}}{8}\].

Note:
It is very important to note that \[\log x\] denotes that the base is \[10\] and \[\ln x\] denotes that base is \[e\]. Also, \[{\log _b}x\] is only defined when \[b\] and \[x\] are two positive real numbers and \[b\] is not equal to \[1\]. Logarithm base \[10\] i.e., \[b = 10\] is called decimal or common logarithm, logarithm base \[e\] is called natural logarithm and binary logarithm uses base \[2\].
Some properties of Logarithmic functions are as follows:
\[(1)\] The Product Rule: \[{\log _b}(mn) = {\log _b}(m) + {\log _b}(n)\]
\[(2)\] The Quotient Rule: \[{\log _b}\dfrac{m}{n} = {\log _b}m - {\log _b}n\]
\[(3)\] The Power Rule: \[{\log _b}\left( {{m^p}} \right) = p{\log _b}m\]
\[(4)\] The Zero Exponent Rule: \[{\log _b}1 = 0\]
\[(5)\] Change of Base Rule: \[{\log _b}x = \dfrac{{{{\log }_a}x}}{{{{\log }_a}b}}\]