2 < 10^x < 5
take the log of all.
log 2< x<log 5
To solve the inequality 2 < 10^x < 5, we can take the logarithm of all sides.
First, let's take the logarithm of the left side, which is 2. We can use the natural logarithm, denoted as ln, or we can use a logarithm with any base, such as logarithm base 10 (log). Let's use the natural logarithm (ln) for this example:
ln(2) < ln(10^x) < ln(5)
Next, we can use the logarithmic property that states the logarithm of a power of a number is equal to the exponent multiplied by the logarithm of the base. In this case, we have:
ln(2) < x * ln(10) < ln(5)
The natural logarithm of 10, ln(10), is approximately 2.3026. Therefore, we can simplify the inequality further:
ln(2) < 2.3026x < ln(5)
To isolate x, we can divide all terms by 2.3026:
ln(2) / 2.3026 < x < ln(5) / 2.3026
Simplifying the expression further, we have:
x > ln(2) / 2.3026 and x < ln(5) / 2.3026
Therefore, the simplified inequality is:
ln(2) / 2.3026 < x < ln(5) / 2.3026