Math Problem Statement

does x log 10^x = x^2

Solution

We are asked if the equation

xlog(10x)=x2x \log(10^x) = x^2

is true. Let's break this down.

  1. Simplifying log(10x)\log(10^x): Recall the logarithmic identity:
    log(ab)=blog(a)\log(a^b) = b \log(a) Applying this to log(10x)\log(10^x), we get: log(10x)=xlog(10)\log(10^x) = x \log(10) Since log(10)=1\log(10) = 1, this simplifies further to: log(10x)=x\log(10^x) = x

  2. Substitute this into the original equation: Now the equation becomes: xx=x2x \cdot x = x^2 Which simplifies to: x2=x2x^2 = x^2

  3. Conclusion: The equation is true for all values of xx. Therefore, xlog(10x)=x2x \log(10^x) = x^2 is indeed true for all xx.

Would you like further clarification on this? Or do you have any other questions?

Here are some related questions to explore further:

  1. What other logarithmic identities are useful in simplifying equations?
  2. How does the behavior of logarithmic and exponential functions compare?
  3. Can we solve equations with logarithms and exponents directly by graphing them?
  4. What are the properties of logarithms that help in simplifying complex equations?
  5. How does the logarithmic function behave as xx approaches 0 or infinity?

Tip: Always remember the key logarithmic identity log(ab)=blog(a)\log(a^b) = b \log(a). It can simplify expressions quickly.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra

Formulas

log(a^b) = b * log(a)
x * x = x^2

Theorems

Logarithmic identities

Suitable Grade Level

Grades 9-11