Proving Logarithmic Equations - College Algebra & Precalculus

TL;DR

Use the Change of Base formula to prove logarithmic equations by making both sides of the equations equal.

Transcript

consider the logarithmic equation shown on a screen how can we prove if this equation is true or not in order to prove this logarithmic equation we need to make either the left side equal to the right side or the right side equal to the left side either way we need to make sure that both sides are exactly the same so what I'm going to do in this ex... Read More

Key Insights

• 👻 The Change of Base formula allows for the comparison and simplification of logarithmic equations.
• ✊ The square root of a can be written as a to the 1/2 power.
• 👻 The property of logarithms allows for the variable to be moved to the front of the logarithm.
• ✖️ Multiplying by 1 does not change the value of a fraction.
• 🧑‍🏭 The value of a fraction can be simplified by multiplying the numerator and denominator by the same factor.
• 🧑‍💻 The Change of Base formula can be used to convert a division of two logs into a single log.
• 🧘 The coefficient or constant in front of a logarithm can be moved to the exponent position.

Questions & Answers

Q: How can we prove the truth of a logarithmic equation?

To prove the truth of a logarithmic equation, we need to make sure that both sides of the equation are equal. This can be done by using the Change of Base formula and simplifying the expression.

Q: What is the Change of Base formula?

The Change of Base formula, log base a of B is equal to log B over log a, allows us to rewrite logarithms with different bases into a common base, making them easier to compare and simplify.

Q: Can we introduce a new base when using the Change of Base formula?

Yes, when using the Change of Base formula, we can introduce a new base, which could be any number, including integers like 5 or 8, or special values like 10 or e. If no base is specified, it is assumed to be 10.

Q: How can we prove the truth of logarithmic equations with exponents?

We can apply the properties of logarithms to move the exponent in front of the logarithm, simplifying the equation. By doing so, we can prove the truth of logarithmic equations with exponents.

Summary & Key Takeaways

• To prove a logarithmic equation, make both sides of the equation equal by using the Change of Base formula.

• The Change of Base formula states that log base a of B is equal to log B over log a.

• Apply the Change of Base formula to rewrite logarithmic equations and simplify them.