Question

How do you solve $\log (x) + \log (x + 3) = 1$?

Answer 1

The above question is based on the concept of logarithms. The main approach towards solving the equation is by applying the logarithm properties. On applying the properties and further on simplifying we will get a quadratic equation where we can find out the value of x.

Complete step by step solution:
Logarithm is the exponent or power to which base must be raised to yield a given number. When expressed mathematically x is the logarithm of base n to the base b if ${b^x} = n$,then we can write it has
$x = {\log _b}n$
Now, assuming that log is referring to base 10 of logarithm. So, now we can applying the logarithm property
$\log (a) + \log (b) = \log (ab)$
So, applying it on the equation we get,

$$\Rightarrow \log \left( x \right) + \log (x + 3) = 1 \\\ \Rightarrow \log \left( {x\left( {x + 3} \right)} \right) = 1 \\\$$

Now by applying another logarithmic property we get,
${10^{\log \left( x \right)}} = x$
Therefore, by applying it on the above step we get,
${10^{\log \left( {x\left( {x - 3} \right)} \right)}} = {10^3}$

$$\Rightarrow x\left( {x + 3} \right) = 10 \\\ \Rightarrow {x^2} + 3x - 10 = 0 \\\$$

We get the above equation as a quadratic equation. So further we will solve it by calculating the factors.
First step is by multiplying the coefficient of ${x^2}$and the constant term -10, we get $- 10{x^2}$.
After this, factors of $- 10{x^2}$ should be calculated in such a way that their addition should be equal to$3x$.
Factors of -10 can be -2 and 5.
where $5x - 2x = 3x$.
So, further we write the equation by equating it with zero and splitting the middle term according to the
factors.

$$\Rightarrow {x^2} + 5x - 2x - 10 = 0 \\\ \Rightarrow x(x + 5) - 2(x + 5) = 0 \\\ \Rightarrow (x - 2)(x + 5) = 0 \\\$$

Therefore, we get the above solution as x=2.

Note: An important thing to note is that since we get two values that are 2 and -5, we consider only the value 2. The reason is -5 excludes the solution that makes $\log (x) + \log (x + 1) = 1$ true. Therefore, The final answer is 5.