Question

How do you solve \[\log x + \log 5 = 2\]?

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,
\[x = {\log _b}y\] 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 be solved by using properties of both logarithms and exponents and 2 is written as \[2{\log _{10}}10\] and then solve the equation to get the required result.

Complete step-by-step solution:
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, For example, the base ten logarithm of 1000 is 3, because ten raised to the power of two is 100:\[\log 1000 = 2\] because \[{10^3} = 1000\]. In general, you write log followed by the base number as a subscript. The most common logarithms are base 10 logarithms and natural logarithms; they have special notations. A base ten log is written as \[\log \], and we use different base units but most common logarithms are base 10 logarithms.
Now given equation is \[\log x + \log 5 = 2\],
Now using logarithmic property \[\log x + \log y = \log xy\], then the equation becomes,
\[\log x + \log 5 = \log 5x\],
Now 2 can be written as,\[2{\log _{10}}10\] then the equation becomes,
\[ \Rightarrow \log 5x = 2{\log _{10}}10\],
Then taking 2 to the power in the logarithms we get,
\[ \Rightarrow \log 5x = {\log _{10}}{10^2}\],
Now taking the square we get,
\[ \Rightarrow \log 5x = \log 100\],
Now equating the terms we get,
\[ \Rightarrow 5x = 100\],
Now dividing both sides with 5 we get,
\[ \Rightarrow \dfrac{{5x}}{5} = \dfrac{{100}}{5}\],
Now simplifying we get,
\[ \Rightarrow x = 20\].
The value of \[x\] is equal to \[20\].

\[\therefore \]The value of \[x\] is equal to \[20\] when \[\log x + \log 5 = 2\] is simplified.

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, in these types of questions, we use logarithmic properties and formulas, and some of useful formulas are:
 \[{\log _a}xy = {\log _a}x + {\log _a}y\],
\[{\log _a}{x^n} = n{\log _a}x\],
\[{\log _a}b = \dfrac{{{{\log }_e}b}}{{{{\log }_e}a}}\],
\[{\log _{\dfrac{1}{a}}}b = - {\log _a}b\],
\[{\log _a}a = 1\],
\[{\log _{{a^x}}}b = \dfrac{1}{x}{\log _a}b\]