$\begin{array}{c}10^x+5=60 \\ x=? ?\end{array}$
Answer (2)
Subtract 5 from both sides of the equation: 1 0 x = 55 .
Take the logarithm base 10 of both sides: lo g 10 ( 1 0 x ) = lo g 10 ( 55 ) .
Simplify using logarithm properties: x = lo g 10 ( 55 ) .
The solution is: lo g 10 ( 55 ) .
Explanation
Problem Analysis We are given the equation 1 0 x + 5 = 60 and we want to find the value of x .
Isolating the Exponential Term First, we need to isolate the term with x . To do this, we subtract 5 from both sides of the equation: 1 0 x + 5 − 5 = 60 − 5
Simplifying the Equation This simplifies to: 1 0 x = 55
Applying Logarithm Now, to solve for x , 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 get: x = lo g 10 ( 55 )
Approximating the Value of x We can approximate the value of lo g 10 ( 55 ) using a calculator. The result is approximately 1.74036.
Final Answer Therefore, the solution for x is lo g 10 ( 55 ) , which is approximately 1.74036.
Examples
Logarithms are incredibly useful in many real-world situations. For example, they are used to measure the magnitude of earthquakes on the Richter scale. The Richter scale is logarithmic, meaning that an increase of one unit on the scale corresponds to a tenfold increase in the amplitude of the earthquake waves. Similarly, logarithms are used in chemistry to measure pH levels, in acoustics to measure sound intensity (decibels), and in finance to calculate compound interest. Understanding logarithms helps us to quantify and analyze phenomena that vary over a wide range of values.
To solve 1 0 x + 5 = 60 , we isolate the exponential term to get 1 0 x = 55 , then take the logarithm base 10 of both sides, yielding x = lo g 10 ( 55 ) , which is approximately 1.74036. This procedure highlights the importance of logarithms in simplifying and solving exponential equations.
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