Sum of logarithms with same base | Logarithms | Algebra II | Khan Academy

TL;DR

Using logarithm properties, we can simplify log base 3 of 27x to log base 3 of 27 plus log base 3 of x.

Transcript

We're asked to simplify log base 3 of 27x. And frankly, this is already quite simple. But I'm assuming they want us to use some logarithm properties and manipulate this in some way, maybe to actually make it a little bit more complicated. But let's give our best shot at it. So the logarithm property that jumps out at me-- because this right over he... Read More

Key Insights

• 😑 Logarithm properties can be used to simplify logarithmic expressions.
• ⚾ The property log base b of a times c equals log base b of a plus log base b of c is derived from exponent properties.
• 🤨 Logarithms represent the exponents necessary to raise a base to a certain value.
• 😑 Simplifying logarithmic expressions can make calculations and manipulations easier.
• 🧑‍💻 Log base 3 of 27 is equal to 3.
• 🧑‍💻 Log base 3 of 27x can be written as 3 plus log base 3 of x using logarithm properties.
• ❓ It is important to understand how to apply logarithm properties before delving into the underlying mathematical intuition.

Questions & Answers

Q: How can we simplify log base 3 of 27x using logarithm properties?

Log base 3 of 27x can be simplified using the property log base b of a times c equals log base b of a plus log base b of c. In this case, we rewrite log base 3 of 27x as log base 3 of 27 plus log base 3 of x.

Q: What is the value of log base 3 of 27?

To find log base 3 of 27, we need to determine what power we have to raise 3 to in order to get 27. Since 3 to the third power equals 27, log base 3 of 27 evaluates to 3.

Q: How do logarithms relate to exponents?

Logarithms are the exponents that we need to raise a base to in order to obtain a certain value. Log base b of a equals x means that b to the x power equals a. Logarithms allow us to solve exponential equations and simplify calculations.

Q: What is the benefit of simplifying log base 3 of 27x?

Simplifying log base 3 of 27x to 3 plus log base 3 of x allows for easier calculations and manipulation of logarithmic expressions. It breaks down a single term into two separate terms, which can be useful in various mathematical operations.

Summary & Key Takeaways

• Log base 3 of 27x can be simplified using the logarithm property log base b of a times c equals log base b of a plus log base b of c.

• By applying this property, log base 3 of 27x becomes log base 3 of 27 plus log base 3 of x.

• The first term, log base 3 of 27, simplifies to 3, leaving us with 3 plus log base 3 of x.