Natural logarithms and exponentials · Short practice

Check a final answer, then explain the reasoning on paper. Guest progress stays in this browser; an account saves future checks without overwriting your first result.

Recall

Natural logarithms and exponentials — objective 1

Maths4U Original · Short practice

Let \(\displaystyle a=\)\(e\), \(p = e^{4}\), \(q = e\). Work exactly, using index laws.

Find \(\displaystyle \log _{a} p\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Fresh review variant

Natural logarithms and exponentials — objective 1

Maths4U Original · Short practice

Let \(\displaystyle a=\)\(e\), \(p = e^{5}\), \(q = e^{2}\). Work exactly, using index laws.

Find \(\displaystyle \log _{a} p\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Method

Natural logarithms and exponentials — objective 2

Maths4U Original · Short practice

Let \(\displaystyle a=\)\(e\), \(p = e^{5}\), \(q = e^{2}\). Work exactly, using index laws.

Find logₐ(pq).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Fresh review variant

Natural logarithms and exponentials — objective 2

Maths4U Original · Short practice

Let \(\displaystyle a=\)\(e\), \(p = e^{6}\), \(q = e^{3}\). Work exactly, using index laws.

Find logₐ(pq).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Apply

Natural logarithms and exponentials — objective 3

Maths4U Original · Short practice

Let \(\displaystyle a=\)\(e\), \(p = e^{6}\), \(q = e^{3}\). Work exactly, using index laws.

Find \(\displaystyle \log_{a} (\frac{p}{q})\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Fresh review variant

Natural logarithms and exponentials — objective 3

Maths4U Original · Short practice

Let \(\displaystyle a=\)\(e\), \(p = e^{3}\), \(q = e\). Work exactly, using index laws.

Find \(\displaystyle \log_{a} (\frac{p}{q})\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Check understanding

Natural logarithms and exponentials — objective 4

Maths4U Original · Short practice

Let \(\displaystyle a=\)\(e\), \(p = e^{7}\), \(q = e^{4}\). Work exactly, using index laws.

Find \(\displaystyle \log_{a} (\frac{p^{2}}{q})\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Fresh review variant

Natural logarithms and exponentials — objective 4

Maths4U Original · Short practice

Let \(\displaystyle a=\)\(e\), \(p = e^{3}\), \(q = e\). Work exactly, using index laws.

Find \(\displaystyle \log_{a} (\frac{p^{2}}{q})\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

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