$\begin{array}{c}10^x+5=60 \\ x=[?]\end{array}$
Answer (2)
Subtract 5 from both sides: 1 0 x = 55 .
Take the logarithm base 10 of both sides: x = lo g 10 ( 55 ) .
Evaluate the logarithm: x ≈ 1.74036 .
The solution is 1.74036 .
Explanation
Understanding the Problem We are given the equation 1 0 x + 5 = 60 and we need to find the value of x .
Isolating the Exponential Term First, we subtract 5 from both sides of the equation to isolate the term with x :
1 0 x + 5 − 5 = 60 − 5
Simplified Equation This simplifies to: 1 0 x = 55
Applying Logarithm Now, we take the logarithm base 10 of both sides of the equation: lo g 10 ( 1 0 x ) = lo g 10 ( 55 )
Solving for x Using the property of logarithms that lo g b ( b x ) = x , we simplify the left side: x = lo g 10 ( 55 )
Final Answer Using a calculator, we find that lo g 10 ( 55 ) ≈ 1.74036 . Therefore, the solution is: x ≈ 1.74036
Examples
Logarithms are incredibly useful in many real-world scenarios. For instance, they are used to measure the intensity of earthquakes on the Richter scale, which is a base-10 logarithmic scale. Similarly, logarithms are used in chemistry to measure pH levels, and in finance to calculate compound interest. Understanding logarithms allows us to work with very large or very small numbers more easily and to model exponential growth and decay.
To solve the equation 1 0 x + 5 = 60 , we first isolate the exponential term to get 1 0 x = 55 . Taking the logarithm base 10 of both sides gives x = lo g 10 ( 55 ) , which evaluates to approximately 1.74036. Thus, the final answer is x ≈ 1.74036 .
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