Without A Calculator Simplify Log6 180

How to Simplify log6 180 Without a Calculator

Simplifying log6 180 involves expressing the logarithm in terms of simpler components using logarithm properties. The key properties we'll use are:

• logₐ(bᶜ) = c
• logₐ(b) + logₐ(c) = logₐ(b × c)
• logₐ(b) - logₐ(c) = logₐ(b/c)

By applying these properties, we can break down log6 180 into simpler logarithmic expressions that are easier to evaluate.

Logarithm Properties Used in Simplification

The fundamental logarithm properties that make simplification possible are:

These properties allow us to convert between different forms of logarithms and simplify complex expressions.

Step-by-Step Simplification Process

Let's simplify log6 180 using these properties:

1. First, express 180 as a product of powers of 6 and other numbers:
   180 = 6 × 30
2. Apply the product rule of logarithms:
   log6 180 = log6 (6 × 30) = log6 6 + log6 30
3. Simplify log6 6 using the power rule:
   log6 6 = 1
4. Now we have:
   log6 180 = 1 + log6 30
5. We can further simplify log6 30 by expressing 30 as a product of powers of 6 and other numbers:
   30 = 6 × 5
6. Apply the product rule again:
   log6 30 = log6 (6 × 5) = log6 6 + log6 5 = 1 + log6 5
7. Substitute back into our previous expression:
   log6 180 = 1 + (1 + log6 5) = 2 + log6 5

Therefore, the simplified form of log6 180 is 2 + log6 5.

Verification of the Simplified Form

To verify our simplified form, we can use the definition of logarithms:

Let's check if 6^(2 + log6 5) equals 180:

This confirms that our simplified form is correct.