Write the equation in logarithmic form.
[tex]$10^v=z$[/tex]
Answer (1)
The problem gives the equation 1 0 v = z .
Apply the definition of logarithm: if b x = y , then lo g b y = x .
Rewrite the equation in logarithmic form: lo g 10 z = v .
The final answer is lo g z = v .
Explanation
Understanding the Problem We are given the equation 1 0 v = z and we want to rewrite it in logarithmic form.
Definition of Logarithm Recall the definition of a logarithm: If b x = y , then the equivalent logarithmic form is lo g b y = x . In other words, the logarithm of y to the base b is x if b raised to the power of x equals y .
Applying the Definition In our case, we have 1 0 v = z . Comparing this with the general form b x = y , we can identify b = 10 , x = v , and y = z . Therefore, we can rewrite the equation in logarithmic form as lo g 10 z = v .
Final Answer Since base 10 logarithms are commonly used, we can also write lo g 10 z as lo g z . Thus, the equation in logarithmic form is lo g z = v .
Examples
Logarithmic equations are used in many real-world applications, such as calculating the magnitude of earthquakes on the Richter scale, measuring the intensity of sound (decibels), and determining the pH of a solution in chemistry. For example, if the intensity of an earthquake is 1000 times greater than the reference intensity, its magnitude on the Richter scale would be lo g 10 ( 1000 ) = 3 . Understanding how to convert between exponential and logarithmic forms allows us to solve these types of problems effectively.
