Question

How do you solve \[3\ln x + \ln 5 = 7\]?

Answer 1

Hint: A logarithm is an exponent which indicates to what power a base must be raised to produce a given number.
 \[y = {b^x}\] exponential form,
\[x = {\ln _b}y\] logarithmic function, where \[x\] is the logarithm of \[y\] to the base \[b\], and \[{\log _b}y\] is the power to which we have to raise \[b\] to get \[y\], we are expressing \[x\] in terms of \[y\].
 Now the given question can solved by using properties of logarithms i.e., \[\ln x + \ln y = \ln \left( {xy} \right)\],\[\ln {x^n} = n\ln x\] and applying the exponents on both sides by using the identity \[{e^{\ln x}} = x\], then solve the expression to get the required result.

Complete step by step solution:
Given equation is \[3\ln x + \ln 5 = 7\],
Now using the logarithmic identity \[\ln {x^n} = n\ln x\], we get,
\[ \Rightarrow \ln {x^3} + \ln 5 = 7\],
Now again applying the logarithmic identity \[\ln x + \ln y = \ln \left( {xy} \right)\], we get,
\[ \Rightarrow \ln 5{x^3} = 7\],
Now applying exponents on both sides we get,
\[ \Rightarrow {e^{\ln 5{x^3}}} = {e^7}\],
Now we know that\[{e^{\ln x}} = x\], we get,
\[ \Rightarrow 5{x^3} = {e^7}\],
Now dividing both sides with 5 we get,
\[ \Rightarrow \dfrac{{5{x^3}}}{5} = \dfrac{{{e^7}}}{5}\],
Now simplifying we get,
\[ \Rightarrow {x^3} = \dfrac{{{e^7}}}{5}\],
Now taking the cube root to the right hand side we get,
\[ \Rightarrow x = {\left( {\dfrac{{{e^7}}}{5}} \right)^{\dfrac{1}{3}}}\],
So, the value of \[x\] is equal to \[{\left( {\dfrac{{{e^7}}}{5}} \right)^{\dfrac{1}{3}}}\].

The value of the given function \[3\ln x + \ln 5 = 7\] is equal to \[{\left( {\dfrac{{{e^7}}}{5}} \right)^{\dfrac{1}{3}}}\].

Note: A logarithm is a mathematical operation that determines how many times a certain number, called the base, is multiplied by itself to reach another number, we know that logarithm is the power to which a number must be raised in order to get some other number, and the base unit is the number being raised to a power. In these types of questions, we use logarithmic properties and formulas, and some of useful formulas are:
 \[{\log _a}xy = {\log _a}x + {\log _a}y\],
\[\log x - \log y = \log \left( {\dfrac{x}{y}} \right)\]
\[{\log _a}{x^n} = n{\log _a}x\],
\[{\log _a}b = \dfrac{{{{\log }_e}b}}{{{{\log }_e}a}}\],
\[{\log _{\dfrac{1}{a}}}b = - {\log _a}b\],
\[{\log _a}a = 1\],
\[{\log _{{a^x}}}b = \dfrac{1}{x}{\log _a}b\].