Math Problem Statement

log4420+(log44)20−(log416)⋅(log464)

Solution

Let's break down the expression and solve it step by step.

You have the expression:

log4420+(log44)20(log416)(log464)\log_4 420 + (\log_4 4) 20 - (\log_4 16) \cdot (\log_4 64)

Step 1: Simplify log44\log_4 4

We know that logbb=1\log_b b = 1 for any base bb. Therefore:

log44=1\log_4 4 = 1

Thus, the term (log44)20(\log_4 4) 20 becomes:

120=201 \cdot 20 = 20

Step 2: Simplify log416\log_4 16

Recall that 16=4216 = 4^2, so we can write:

log416=log4(42)=2\log_4 16 = \log_4 (4^2) = 2

Step 3: Simplify log464\log_4 64

Similarly, 64=4364 = 4^3, so:

log464=log4(43)=3\log_4 64 = \log_4 (4^3) = 3

Step 4: Substitute back into the expression

Now, the expression becomes:

log4420+20(23)\log_4 420 + 20 - (2 \cdot 3)

=log4420+206= \log_4 420 + 20 - 6

=log4420+14= \log_4 420 + 14

Step 5: Final Expression

Thus, the simplified expression is:

log4420+14\log_4 420 + 14

Now, unless you want to approximate log4420\log_4 420, this is the simplified form. Would you like me to approximate log4420\log_4 420 or provide further details?


Related questions:

  1. How do you simplify logarithms with different bases?
  2. What are properties of logarithms that can help simplify expressions?
  3. How do you convert logarithms to different bases (e.g., from base 4 to base 10)?
  4. How do you simplify logarithmic expressions with exponents?
  5. Can you solve logarithmic equations by substitution?

Tip:

Remember that logarithms with the same base can be combined using the product, quotient, and power rules.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Properties of Logarithms
Logarithmic Simplification

Formulas

log_b(b) = 1
log_b(a^n) = n * log_b(a)

Theorems

Logarithm Product Rule
Logarithm Quotient Rule

Suitable Grade Level

Grades 9-11